2013

• Paper 1, Section I, D

  #Analysis I

  2013

  Show that $\exp (x) \geqslant 1+x$ for $x \geqslant 0$.

  Let $\left(a_{j}\right)$ be a sequence of positive real numbers. Show that for every $n$,

  $$\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)$$

  Deduce that $\prod_{1}^{n}\left(1+a_{j}\right)$ tends to a limit as $n \rightarrow \infty$ if and only if $\sum_{1}^{n} a_{j}$ does.

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• Paper 1, Section I, F

  #Analysis I

  2013

  (a) Suppose $b_{n} \geqslant b_{n+1} \geqslant 0$ for $n \geqslant 1$ and $b_{n} \rightarrow 0$. Show that $\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}$ converges.

  (b) Does the series $\sum_{n=2}^{\infty} \frac{1}{n \log n}$ converge or diverge? Explain your answer.

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• Paper 1, Section II, D

  #Analysis I

  2013

  (a) Determine the radius of convergence of each of the following power series:

  $$\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}$$

  (b) State Taylor's theorem.

  Show that

  $$(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}$$

  for all $x \in(0,1)$, where

  $$c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}$$

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• Paper 1, Section II, E

  #Analysis I

  2013

  (a) Let $f:[a, b] \rightarrow \mathbb{R}$. Suppose that for every sequence $\left(x_{n}\right)$ in $[a, b]$ with limit $y \in[a, b]$, the sequence $\left(f\left(x_{n}\right)\right)$ converges to $f(y)$. Show that $f$ is continuous at $y$.

  (b) State the Intermediate Value Theorem.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a function with $f(a)=c<f(b)=d$. We say $f$ is injective if for all $x, y \in[a, b]$ with $x \neq y$, we have $f(x) \neq f(y)$. We say $f$ is strictly increasing if for all $x, y$ with $x<y$, we have $f(x)<f(y)$.

  (i) Suppose $f$ is strictly increasing. Show that it is injective, and that if $f(x)<f(y)$ then $x<y .$

  (ii) Suppose $f$ is continuous and injective. Show that if $a<x<b$ then $c<f(x)<d$. Deduce that $f$ is strictly increasing.

  (iii) Suppose $f$ is strictly increasing, and that for every $y \in[c, d]$ there exists $x \in[a, b]$ with $f(x)=y$. Show that $f$ is continuous at $b$. Deduce that $f$ is continuous on $[a, b]$.

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• Paper 1, Section II, E

  #Analysis I

  2013

  (i) State (without proof) Rolle's Theorem.

  (ii) State and prove the Mean Value Theorem.

  (iii) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$ with $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$. Show that there exists $\xi \in(a, b)$ such that

  $$\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$

  Deduce that if moreover $f(a)=g(a)=0$, and the limit

  $$\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$$

  exists, then

  $$\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a$$

  (iv) Deduce that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable then for any $a \in \mathbb{R}$

  $$f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .$$

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• Paper 1, Section II, F

  #Analysis I

  2013

  Fix a closed interval $[a, b]$. For a bounded function $f$ on $[a, b]$ and a dissection $\mathcal{D}$ of $[a, b]$, how are the lower sum $s(f, \mathcal{D})$ and upper sum $S(f, \mathcal{D})$ defined? Show that $s(f, \mathcal{D}) \leqslant S(f, \mathcal{D})$.

  Suppose $\mathcal{D}^{\prime}$ is a dissection of $[a, b]$ such that $\mathcal{D} \subseteq \mathcal{D}^{\prime}$. Show that

  $$s(f, \mathcal{D}) \leqslant s\left(f, \mathcal{D}^{\prime}\right) \text { and } S\left(f, \mathcal{D}^{\prime}\right) \leqslant S(f, \mathcal{D})$$

  By using the above inequalities or otherwise, show that if $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are two dissections of $[a, b]$ then

  $$s\left(f, \mathcal{D}_{1}\right) \leqslant S\left(f, \mathcal{D}_{2}\right)$$

  For a function $f$ and dissection $\mathcal{D}=\left\{x_{0}, \ldots, x_{n}\right\}$ let

  $$p(f, \mathcal{D})=\prod_{k=1}^{n}\left[1+\left(x_{k}-x_{k-1}\right) \inf _{x \in\left[x_{k-1}, x_{k}\right]} f(x)\right]$$

  If $f$ is non-negative and Riemann integrable, show that

  $$p(f, \mathcal{D}) \leqslant e^{\int_{a}^{b} f(x) d x} .$$

  [You may use without proof the inequality $e^{t} \geqslant t+1$ for all $t$.]

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• Paper 2, Section I, A

  #Differential Equations

  2013

  Solve the equation

  $$\ddot{y}-\dot{y}-2 y=3 e^{2 t}+3 e^{-t}+3+6 t$$

  subject to the conditions $y=\dot{y}=0$ at $t=0$.

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• Paper 2, Section I, A

  #Differential Equations

  2013

  Use the transformation $z=\ln x$ to solve

  $$\ddot{z}=-\dot{z}^{2}-1-e^{-z}$$

  subject to the conditions $z=0$ and $\dot{z}=V$ at $t=0$, where $V$ is a positive constant.

  Show that when $\dot{z}(t)=0$

  $$z=\ln \left(\sqrt{V^{2}+4}-1\right)$$

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• Paper 2, Section II, A

  #Differential Equations

  2013

  The function $y(x)$ satisfies the equation

  $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0 .$$

  Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

  For the equation

  $$x y^{\prime \prime}+y=0$$

  classify the point $x=0$ according to your definitions. Find the series solution about $x=0$ which satisfies

  $$y=0 \quad \text { and } \quad y^{\prime}=1 \quad \text { at } x=0$$

  For a second solution with $y=1$ at $x=0$, consider an expansion

  $$y(x)=y_{0}(x)+y_{1}(x)+y_{2}(x)+\ldots,$$

  where $y_{0}=1$ and $x y_{n+1}^{\prime \prime}=-y_{n}$. Find $y_{1}$ and $y_{2}$ which have $y_{n}(0)=0$ and $y_{n}^{\prime}(1)=0$. Comment on $y^{\prime}$ near $x=0$ for this second solution.

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• Paper 2, Section II, A

  #Differential Equations

  2013

  Find $x(t)$ and $y(t)$ which satisfy

  $$\begin{aligned} &3 \dot{x}+\dot{y}+5 x-y=2 e^{-t}+4 e^{-3 t} \\ &\dot{x}+4 \dot{y}-2 x+7 y=-3 e^{-t}+5 e^{-3 t} \end{aligned}$$

  subject to $x=y=0$ at $t=0$.

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• Paper 2, Section II, A

  #Differential Equations

  2013

  Medical equipment is sterilised by placing it in a hot oven for a time $T$ and then removing it and letting it cool for the same time. The equipment at temperature $\theta(t)$ warms and cools at a rate equal to the product of a constant $\alpha$ and the difference between its temperature and its surroundings, $\theta_{1}$ when warming in the oven and $\theta_{0}$ when cooling outside. The equipment starts the sterilisation process at temperature $\theta_{0}$.

  Bacteria are killed by the heat treatment. Their number $N(t)$ decreases at a rate equal to the product of the current number and a destruction factor $\beta$. This destruction factor varies linearly with temperature, vanishing at $\theta_{0}$ and having a maximum $\beta_{\max }$ at $\theta_{1}$.

  Find an implicit equation for $T$ such that the number of bacteria is reduced by a factor of $10^{-20}$ by the sterilisation process.

  A second hardier species of bacteria requires the oven temperature to be increased to achieve the same destruction factor $\beta_{\max }$. How is the sterilisation time $T$ affected?

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• Paper 2, Section II, $6 \mathrm{~A}$

  #Differential Equations

  2013

  Consider the function

  $$f(x, y)=\left(x^{2}-y^{4}\right)\left(1-x^{2}-y^{4}\right)$$

  Determine the type of each of the nine critical points.

  Sketch contours of constant $f(x, y)$.

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• Paper 4, Section I, B

  #Dynamics and Relativity

  2013

  A hot air balloon of mass $M$ is equipped with a bag of sand of mass $m=m(t)$ which decreases in time as the sand is gradually released. In addition to gravity the balloon experiences a constant upwards buoyancy force $T$ and we neglect air resistance effects. Show that if $v(t)$ is the upward speed of the balloon then

  $$(M+m) \frac{d v}{d t}=T-(M+m) g .$$

  Initially at $t=0$ the mass of sand is $m(0)=m_{0}$ and the balloon is at rest in equilibrium. Subsequently the sand is released at a constant rate and is depleted in a time $t_{0}$. Show that the speed of the balloon at time $t_{0}$ is

  $$g t_{0}\left(\left(1+\frac{M}{m_{0}}\right) \ln \left(1+\frac{m_{0}}{M}\right)-1\right)$$

  [You may use without proof the indefinite integral $\int t /(A-t) d t=-t-A \ln |A-t|+C .$ ]

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• Paper 4, Section $I$, B

  #Dynamics and Relativity

  2013

  A frame $S^{\prime}$ moves with constant velocity $v$ along the $x$ axis of an inertial frame $S$ of Minkowski space. A particle $P$ moves with constant velocity $u^{\prime}$ along the $x^{\prime}$ axis of $S^{\prime}$. Find the velocity $u$ of $P$ in $S$.

  The rapidity $\varphi$ of any velocity $w$ is defined by $\tanh \varphi=w / c$. Find a relation between the rapidities of $u, u^{\prime}$ and $v$.

  Suppose now that $P$ is initially at rest in $S$ and is subsequently given $n$ successive velocity increments of $c / 2$ (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of $P$ in $S$ is

  $$c\left(\frac{e^{2 n \alpha}-1}{e^{2 n \alpha}+1}\right)$$

  where $\tanh \alpha=1 / 2$.

  [You may use without proof the addition formulae $\sinh (a+b)=\sinh a \cosh b+\cosh a \sinh b$ and $\cosh (a+b)=\cosh a \cosh b+\sinh a \sinh b$.]

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2013

  (a) A particle $P$ of unit mass moves in a plane with polar coordinates $(r, \theta)$. You may assume that the radial and angular components of the acceleration are given by $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$, where the dot denotes $d / d t$. The particle experiences a central force corresponding to a potential $V=V(r)$.

  (i) Prove that $l=r^{2} \dot{\theta}$ is constant in time and show that the time dependence of the radial coordinate $r(t)$ is equivalent to the motion of a particle in one dimension $x$ in a potential $V_{\text {eff }}$ given by

  $$V_{\text {eff }}=V(x)+\frac{l^{2}}{2 x^{2}}$$

  (ii) Now suppose that $V(r)=-e^{-r}$. Show that if $l^{2}<27 / e^{3}$ then two circular orbits are possible with radii $r_{1}<3$ and $r_{2}>3$. Determine whether each orbit is stable or unstable.

  (b) Kepler's first and second laws for planetary motion are the following statements:

  K1: the planet moves on an ellipse with a focus at the Sun;

  K2: the line between the planet and the Sun sweeps out equal areas in equal times.

  Show that K2 implies that the force acting on the planet is a central force.

  Show that K2 together with $\mathbf{K 1}$ implies that the force is given by the inverse square law.

  [You may assume that an ellipse with a focus at the origin has polar equation $\frac{A}{r}=1+\varepsilon \cos \theta$ with $A>0$ and $0<\varepsilon<1$.]

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2013

  (a) A rigid body $Q$ is made up of $N$ particles of masses $m_{i}$ at positions $\mathbf{r}_{i}(t)$. Let $\mathbf{R}(t)$ denote the position of its centre of mass. Show that the total kinetic energy of $Q$ may be decomposed into $T_{1}$, the kinetic energy of the centre of mass, plus a term $T_{2}$ representing the kinetic energy about the centre of mass.

  Suppose now that $Q$ is rotating with angular velocity $\boldsymbol{\omega}$ about its centre of mass. Define the moment of inertia $I$ of $Q$ (about the axis defined by $\boldsymbol{\omega}$ ) and derive an expression for $T_{2}$ in terms of $I$ and $\omega=|\omega|$.

  (b) Consider a uniform rod $A B$ of length $2 l$ and mass $M$. Two such rods $A B$ and $B C$ are freely hinged together at $B$. The end $A$ is attached to a fixed point $O$ on a perfectly smooth horizontal floor and $A B$ is able to rotate freely about $O$. The rods are initially at rest, lying in a vertical plane with $C$ resting on the floor and each rod making angle $\alpha$ with the horizontal. The rods subsequently move under gravity in their vertical plane.

  Find an expression for the angular velocity of rod $A B$ when it makes angle $\theta$ with the floor. Determine the speed at which the hinge strikes the floor.

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• Paper 4 , Section II, B

  #Dynamics and Relativity

  2013

  (i) An inertial frame $S$ has orthonormal coordinate basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. A second frame $S^{\prime}$ rotates with angular velocity $\boldsymbol{\omega}$ relative to $S$ and has coordinate basis vectors $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$. The motion of $S^{\prime}$ is characterised by the equations $d \mathbf{e}_{i}^{\prime} / d t=\boldsymbol{\omega} \times \mathbf{e}_{i}^{\prime}$ and at $t=0$ the two coordinate frames coincide.

  If a particle $P$ has position vector $\mathbf{r}$ show that $\mathbf{v}=\mathbf{v}^{\prime}+\boldsymbol{\omega} \times \mathbf{r}$ where $\mathbf{v}$ and $\mathbf{v}^{\prime}$ are the velocity vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$.

  (ii) For the remainder of this question you may assume that $\mathbf{a}=\mathbf{a}^{\prime}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$ where $\mathbf{a}$ and $\mathbf{a}^{\prime}$ are the acceleration vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$, and that $\omega$ is constant.

  Consider again the frames $S$ and $S^{\prime}$ in (i). Suppose that $\omega=\omega \mathbf{e}_{3}$ with $\omega$ constant. A particle of mass $m$ moves under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{r}$. When viewed in $S^{\prime}$ its position and velocity at time $t=0$ are $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(1,0,0)$ and $\left(\dot{x}^{\prime}, \dot{y}^{\prime}, \dot{z}^{\prime}\right)=(0,0,0)$. Find the motion of the particle in the coordinates of $S^{\prime}$. Show that for an observer fixed in $S^{\prime}$, the particle achieves its maximum speed at time $t=\pi /(4 \omega)$ and determine that speed. [Hint: you may find it useful to consider the combination $\zeta=x^{\prime}+i y^{\prime}$.]

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2013

  (a) Let $S$ with coordinates $(c t, x, y)$ and $S^{\prime}$ with coordinates $\left(c t^{\prime}, x^{\prime}, y^{\prime}\right)$ be inertial frames in Minkowski space with two spatial dimensions. $S^{\prime}$ moves with velocity $v$ along the $x$-axis of $S$ and they are related by the standard Lorentz transformation:

  $$\left(\begin{array}{c} c t \\ x \\ y \end{array}\right)=\left(\begin{array}{ccc} \gamma & \gamma v / c & 0 \\ \gamma v / c & \gamma & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} c t^{\prime} \\ x^{\prime} \\ y^{\prime} \end{array}\right), \quad \text { where } \gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}} .$$

  A photon is emitted at the spacetime origin. In $S^{\prime}$ it has frequency $\nu^{\prime}$ and propagates at angle $\theta^{\prime}$ to the $x^{\prime}$-axis.

  Write down the 4 -momentum of the photon in the frame $S^{\prime}$.

  Hence or otherwise find the frequency of the photon as seen in $S$. Show that it propagates at angle $\theta$ to the $x$-axis in $S$, where

  $$\tan \theta=\frac{\tan \theta^{\prime}}{\gamma\left(1+\frac{v}{c} \sec \theta^{\prime}\right)}$$

  A light source in $S^{\prime}$ emits photons uniformly in all directions in the $x^{\prime} y^{\prime}$-plane. Show that for large $v$, in $S$ half of the light is concentrated into a narrow cone whose semi-angle $\alpha$ is given by $\cos \alpha=v / c$.

  (b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.

  Two particles $A_{1}$ and $A_{2}$ of rest masses $m_{1}$ and $m_{2}$ move collinearly with uniform velocities $u_{1}$ and $u_{2}$ respectively, along the $x$-axis of a frame $S$. They collide, coalescing to form a single particle $A_{3}$.

  Determine the velocity of the centre-of-mass frame of the system comprising $A_{1}$ and $A_{2}$.

  Find the speed of $A_{3}$ in $S$ and show that its rest mass $m_{3}$ is given by

  $$m_{3}^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1} \gamma_{2}\left(1-\frac{u_{1} u_{2}}{c^{2}}\right),$$

  where $\gamma_{i}=\left(1-u_{i}^{2} / c^{2}\right)^{-1 / 2}$

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• Paper 3, Section I, D

  #Groups

  2013

  State Lagrange's Theorem.

  Let $G$ be a finite group, and $H$ and $K$ two subgroups of $G$ such that

  (i) the orders of $H$ and $K$ are coprime;

  (ii) every element of $G$ may be written as a product $h k$, with $h \in H$ and $k \in K$;

  (iii) both $H$ and $K$ are normal subgroups of $G$.

  Prove that $G$ is isomorphic to $H \times K$.

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• Paper 3, Section I, D

  #Groups

  2013

  Define what it means for a group to be cyclic, and for a group to be abelian. Show that every cyclic group is abelian, and give an example to show that the converse is false.

  Show that a group homomorphism from the cyclic group $C_{n}$ of order $n$ to a group $G$ determines, and is determined by, an element $g$ of $G$ such that $g^{n}=1$.

  Hence list all group homomorphisms from $C_{4}$ to the symmetric group $S_{4}$.

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• Paper 3, Section II, D

  #Groups

  2013

  (a) Let $G$ be a finite group. Show that there exists an injective homomorphism $G \rightarrow \operatorname{Sym}(X)$ to a symmetric group, for some set $X$.

  (b) Let $H$ be the full group of symmetries of the cube, and $X$ the set of edges of the cube.

  Show that $H$ acts transitively on $X$, and determine the stabiliser of an element of $X$. Hence determine the order of $H$.

  Show that the action of $H$ on $X$ defines an injective homomorphism $H \rightarrow \operatorname{Sym}(X)$ to the group of permutations of $X$, and determine the number of cosets of $H$ in $\operatorname{Sym}(X)$.

  Is $H$ a normal subgroup of $\operatorname{Sym}(X) ?$ Prove your answer.

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• Paper 3, Section II, D

  #Groups

  2013

  (a) Let $p$ be a prime, and let $G=S L_{2}(p)$ be the group of $2 \times 2$ matrices of determinant 1 with entries in the field $\mathbb{F}_{p}$ of integers $\bmod p$.

  (i) Define the action of $G$ on $X=\mathbb{F}_{p} \cup\{\infty\}$ by Möbius transformations. [You need not show that it is a group action.]

  State the orbit-stabiliser theorem.

  Determine the orbit of $\infty$ and the stabiliser of $\infty$. Hence compute the order of $S L_{2}(p)$.

  (ii) Let

  $$A=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right), \quad B=\left(\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right)$$

  Show that $A$ is conjugate to $B$ in $G$ if $p=11$, but not if $p=5$.

  (b) Let $G$ be the set of all $3 \times 3$ matrices of the form

  $$\left(\begin{array}{lll} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)$$

  where $a, b, x \in \mathbb{R}$. Show that $G$ is a subgroup of the group of all invertible real matrices.

  Let $H$ be the subset of $G$ given by matrices with $a=0$. Show that $H$ is a normal subgroup, and that the quotient group $G / H$ is isomorphic to $\mathbb{R}$.

  Determine the centre $Z(G)$ of $G$, and identify the quotient group $G / Z(G)$.

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• Paper 3, Section II, D

  #Groups

  2013

  (a) Let $G$ be the dihedral group of order $4 n$, the symmetry group of a regular polygon with $2 n$ sides.

  Determine all elements of order 2 in $G$. For each element of order 2 , determine its conjugacy class and the smallest normal subgroup containing it.

  (b) Let $G$ be a finite group.

  (i) Prove that if $H$ and $K$ are subgroups of $G$, then $K \cup H$ is a subgroup if and only if $H \subseteq K$ or $K \subseteq H$.

  (ii) Let $H$ be a proper subgroup of $G$, and write $G \backslash H$ for the elements of $G$ not in $H$. Let $K$ be the subgroup of $G$ generated by $G \backslash H$.

  Show that $K=G$.

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• Paper 3, Section II, D

  #Groups

  2013

  Let $p$ be a prime number.

  Prove that every group whose order is a power of $p$ has a non-trivial centre.

  Show that every group of order $p^{2}$ is abelian, and that there are precisely two of them, up to isomorphism.

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• Paper 4, Section I, E

  #Numbers and Sets

  2013

  Let $m$ and $n$ be positive integers. State what is meant by the greatest common divisor $\operatorname{gcd}(m, n)$ of $m$ and $n$, and show that there exist integers $a$ and $b$ such that $\operatorname{gcd}(m, n)=a m+b n$. Deduce that an integer $k$ divides both $m$ and $n$ only if $k$ divides $\operatorname{gcd}(m, n)$.

  Prove (without using the Fundamental Theorem of Arithmetic) that for any positive integer $k, \operatorname{gcd}(k m, k n)=k \operatorname{gcd}(m, n)$.

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• Paper 4, Section I, E

  #Numbers and Sets

  2013

  Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a sequence of real numbers. What does it mean to say that the sequence $\left(x_{n}\right)$ is convergent? What does it mean to say the series $\sum x_{n}$ is convergent? Show that if $\sum x_{n}$ is convergent, then the sequence $\left(x_{n}\right)$ converges to zero. Show that the converse is not necessarily true.

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• Paper 4, Section II, E

  #Numbers and Sets

  2013

  (i) What does it mean to say that a function $f: X \rightarrow Y$ is injective? What does it mean to say that $f$ is surjective? Let $g: Y \rightarrow Z$ be a function. Show that if $g \circ f$ is injective, then so is $f$, and that if $g \circ f$ is surjective, then so is $g$.

  (ii) Let $X_{1}, X_{2}$ be two sets. Their product $X_{1} \times X_{2}$ is the set of ordered pairs $\left(x_{1}, x_{2}\right)$ with $x_{i} \in X_{i}(i=1,2)$. Let $p_{i}$ (for $\left.i=1,2\right)$ be the function

  $$p_{i}: X_{1} \times X_{2} \rightarrow X_{i}, \quad p_{i}\left(x_{1}, x_{2}\right)=x_{i}$$

  When is $p_{i}$ surjective? When is $p_{i}$ injective?

  (iii) Now let $Y$ be any set, and let $f_{1}: Y \rightarrow X_{1}, f_{2}: Y \rightarrow X_{2}$ be functions. Show that there exists a unique $g: Y \rightarrow X_{1} \times X_{2}$ such that $f_{1}=p_{1} \circ g$ and $f_{2}=p_{2} \circ g$.

  Show that if $f_{1}$ or $f_{2}$ is injective, then $g$ is injective. Is the converse true? Justify your answer.

  Show that if $g$ is surjective then both $f_{1}$ and $f_{2}$ are surjective. Is the converse true? Justify your answer.

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• Paper 4, Section II, E

  #Numbers and Sets

  2013

  (i) Let $N$ and $r$ be integers with $N \geqslant 0, r \geqslant 1$. Let $S$ be the set of $(r+1)$-tuples $\left(n_{0}, n_{1}, \ldots, n_{r}\right)$ of non-negative integers satisfying the equation $n_{0}+\cdots+n_{r}=N$. By mapping elements of $S$ to suitable subsets of $\{1, \ldots, N+r\}$ of size $r$, or otherwise, show that the number of elements of $S$ equals

  $$\left(\begin{array}{c} N+r \\ r \end{array}\right)$$

  (ii) State the Inclusion-Exclusion principle.

  (iii) Let $a_{0}, \ldots, a_{r}$ be positive integers. Show that the number of $(r+1)$-tuples $\left(n_{i}\right)$ of integers satisfying

  $$n_{0}+\cdots+n_{r}=N, \quad 0 \leqslant n_{i}<a_{i} \text { for all } i$$

  $$\begin{aligned} \left(\begin{array}{c} N+r \\ r \end{array}\right) &-\sum_{0 \leqslant i \leqslant r}\left(\begin{array}{c} N+r-a_{i} \\ r \end{array}\right)+\sum_{0 \leqslant i<j \leqslant r}\left(\begin{array}{c} N+r-a_{i}-a_{j} \\ r \end{array}\right) \\ &-\sum_{0 \leqslant i<j<k \leqslant r}\left(\begin{array}{c} N+r-a_{i}-a_{j}-a_{k} \\ r \end{array}\right)+\cdots \end{aligned}$$

  where the binomial coefficient $\left(\begin{array}{c}m \\ r\end{array}\right)$ is defined to be zero if $m<r$.

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• Paper 4, Section II, $7 \mathrm{E}$

  #Numbers and Sets

  2013

  (i) What does it mean to say that a set is countable? Show directly from your definition that any subset of a countable set is countable, and that a countable union of countable sets is countable.

  (ii) Let $X$ be either $\mathbb{Z}$ or $\mathbb{Q}$. A function $f: X \rightarrow \mathbb{Z}$ is said to be periodic if there exists a positive integer $n$ such that for every $x \in X, f(x+n)=f(x)$. Show that the set of periodic functions from $\mathbb{Z}$ to itself is countable. Is the set of periodic functions $f: \mathbb{Q} \rightarrow \mathbb{Z}$ countable? Justify your answer.

  (iii) Show that $\mathbb{R}^{2}$ is not the union of a countable collection of lines.

  [You may assume that $\mathbb{R}$ and the power set of $\mathbb{N}$ are uncountable.]

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• Paper 4, Section II, E

  #Numbers and Sets

  2013

  Let $p$ be a prime number, and $x, n$ integers with $n \geqslant 1$.

  (i) Prove Fermat's Little Theorem: for any integer $x, x^{p} \equiv x(\bmod p)$.

  (ii) Show that if $y$ is an integer such that $x \equiv y\left(\bmod p^{n}\right)$, then for every integer $r \geqslant 0$,

  $$x^{p^{r}} \equiv y^{p^{r}}\left(\bmod p^{n+r}\right)$$

  Deduce that $x^{p^{n}} \equiv x^{p^{n-1}}\left(\bmod p^{n}\right) .$

  (iii) Show that there exists a unique integer $y \in\left\{0,1, \ldots, p^{n}-1\right\}$ such that

  $$y \equiv x \quad(\bmod p) \quad \text { and } \quad y^{p} \equiv y \quad\left(\bmod p^{n}\right)$$

  comment

• Paper 2, Section I, F

  #Probability

  2013

  Let $X$ be a random variable with mean $\mu$ and variance $\sigma^{2}$. Let

  $$G(a)=\mathbb{E}\left[(X-a)^{2}\right]$$

  Show that $G(a) \geqslant \sigma^{2}$ for all $a$. For what value of $a$ is there equality?

  Let

  $$H(a)=\mathbb{E}[|X-a|]$$

  Supposing that $X$ has probability density function $f$, express $H(a)$ in terms of $f$. Show that $H$ is minimised when $a$ is such that $\int_{-\infty}^{a} f(x) d x=1 / 2$.

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• Paper 2, Section I, F

  #Probability

  2013

  (i) Let $X$ be a random variable. Use Markov's inequality to show that

  $$\mathbb{P}(X \geqslant k) \leqslant \mathbb{E}\left(e^{t X}\right) e^{-k t}$$

  for all $t \geqslant 0$ and real $k$.

  (ii) Calculate $\mathbb{E}\left(e^{t X}\right)$ in the case where $X$ is a Poisson random variable with parameter $\lambda=1$. Using the inequality from part (i) with a suitable choice of $t$, prove that

  $$\frac{1}{k !}+\frac{1}{(k+1) !}+\frac{1}{(k+2) !}+\ldots \leqslant\left(\frac{e}{k}\right)^{k}$$

  for all $k \geqslant 1$.

  comment

• Paper 2, Section II, F

  #Probability

  2013

  Let $Z$ be an exponential random variable with parameter $\lambda=1$. Show that

  $$\mathbb{P}(Z>s+t \mid Z>s)=\mathbb{P}(Z>t)$$

  for any $s, t \geqslant 0$.

  Let $Z_{\text {int }}=\lfloor Z\rfloor$ be the greatest integer less than or equal to $Z$. What is the probability mass function of $Z_{\text {int }}$ ? Show that $\mathbb{E}\left(Z_{\text {int }}\right)=\frac{1}{e-1}$.

  Let $Z_{\mathrm{frac}}=Z-Z_{\mathrm{int}}$ be the fractional part of $Z$. What is the density of $Z_{\mathrm{frac}}$ ?

  Show that $Z_{\text {int }}$ and $Z_{\text {frac }}$ are independent.

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• Paper 2, Section II, F

  #Probability

  2013

  Let $X$ be a random variable taking values in the non-negative integers, and let $G$ be the probability generating function of $X$. Assuming $G$ is everywhere finite, show that

  $$G^{\prime}(1)=\mu \text { and } G^{\prime \prime}(1)=\sigma^{2}+\mu^{2}-\mu$$

  where $\mu$ is the mean of $X$ and $\sigma^{2}$ is its variance. [You may interchange differentiation and expectation without justification.]

  Consider a branching process where individuals produce independent random numbers of offspring with the same distribution as $X$. Let $X_{n}$ be the number of individuals in the $n$-th generation, and let $G_{n}$ be the probability generating function of $X_{n}$. Explain carefully why

  $$G_{n+1}(t)=G_{n}(G(t))$$

  Assuming $X_{0}=1$, compute the mean of $X_{n}$. Show that

  $$\operatorname{Var}\left(X_{n}\right)=\sigma^{2} \frac{\mu^{n-1}\left(\mu^{n}-1\right)}{\mu-1}$$

  Suppose $\mathbb{P}(X=0)=3 / 7$ and $\mathbb{P}(X=3)=4 / 7$. Compute the probability that the population will eventually become extinct. You may use standard results on branching processes as long as they are clearly stated.

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• Paper 2, Section II, F

  #Probability

  2013

  Let $X$ be a geometric random variable with $\mathbb{P}(X=1)=p$. Derive formulae for $\mathbb{E}(X)$ and $\operatorname{Var}(X)$ in terms of $p .$

  A jar contains $n$ balls. Initially, all of the balls are red. Every minute, a ball is drawn at random from the jar, and then replaced with a green ball. Let $T$ be the number of minutes until the jar contains only green balls. Show that the expected value of $T$ is $n \sum_{i=1}^{n} 1 / i$. What is the variance of $T ?$

  comment

• Paper 2, Section II, F

  #Probability

  2013

  Let $\Omega$ be the sample space of a probabilistic experiment, and suppose that the sets $B_{1}, B_{2}, \ldots, B_{k}$ are a partition of $\Omega$ into events of positive probability. Show that

  $$\mathbb{P}\left(B_{i} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)}{\sum_{j=1}^{k} \mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}$$

  for any event $A$ of positive probability.

  A drawer contains two coins. One is an unbiased coin, which when tossed, is equally likely to turn up heads or tails. The other is a biased coin, which will turn up heads with probability $p$ and tails with probability $1-p$. One coin is selected (uniformly) at random from the drawer. Two experiments are performed:

  (a) The selected coin is tossed $n$ times. Given that the coin turns up heads $k$ times and tails $n-k$ times, what is the probability that the coin is biased?

  (b) The selected coin is tossed repeatedly until it turns up heads $k$ times. Given that the coin is tossed $n$ times in total, what is the probability that the coin is biased?

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• Paper 3, Section I, C

  #Vector Calculus

  2013

  The curve $C$ is given by

  $$\mathbf{r}(t)=\left(\sqrt{2} e^{t},-e^{t} \sin t, e^{t} \cos t\right), \quad-\infty<t<\infty$$

  (i) Compute the arc length of $C$ between the points with $t=0$ and $t=1$.

  (ii) Derive an expression for the curvature of $C$ as a function of arc length $s$ measured from the point with $t=0$.

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• Paper 3, Section $I$, C

  #Vector Calculus

  2013

  State a necessary and sufficient condition for a vector field $\mathbf{F}$ on $\mathbb{R}^{3}$ to be conservative.

  Check that the field

  $$\mathbf{F}=\left(2 x \cos y-2 z^{3}, 3+2 y e^{z}-x^{2} \sin y, y^{2} e^{z}-6 x z^{2}\right)$$

  is conservative and find a scalar potential for $\mathbf{F}$.

  comment

• Paper 3, Section II, C

  #Vector Calculus

  2013

  Give an explicit formula for $\mathcal{J}$ which makes the following result hold:

  $$\int_{D} f d x d y d z=\int_{D^{\prime}} \phi|\mathcal{J}| d u d v d w$$

  where the region $D$, with coordinates $x, y, z$, and the region $D^{\prime}$, with coordinates $u, v, w$, are in one-to-one correspondence, and

  $$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$$

  Explain, in outline, why this result holds.

  Let $D$ be the region in $\mathbb{R}^{3}$ defined by $4 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9$ and $z \geqslant 0$. Sketch the region and employ a suitable transformation to evaluate the integral

  $$\int_{D}\left(x^{2}+y^{2}\right) d x d y d z$$

  comment

• Paper 3, Section II, C

  #Vector Calculus

  2013

  Consider the bounded surface $S$ that is the union of $x^{2}+y^{2}=4$ for $-2 \leqslant z \leqslant 2$ and $(4-z)^{2}=x^{2}+y^{2}$ for $2 \leqslant z \leqslant 4$. Sketch the surface.

  Using suitable parametrisations for the two parts of $S$, calculate the integral

  $$\int_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}$$

  for $\mathbf{F}=y z^{2} \mathbf{i}$.

  Check your result using Stokes's Theorem.

  comment

• Paper 3, Section II, C

  #Vector Calculus

  2013

  If $\mathbf{E}$ and $\mathbf{B}$ are vectors in $\mathbb{R}^{3}$, show that

  $$T_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)$$

  is a second rank tensor.

  Now assume that $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey Maxwell's equations, which in suitable units read

  $$\begin{aligned} &\nabla \cdot \mathbf{E}=\rho \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$

  where $\rho$ is the charge density and $\mathbf{J}$ the current density. Show that

  $$\frac{\partial}{\partial t}(\mathbf{E} \times \mathbf{B})=\mathbf{M}-\rho \mathbf{E}-\mathbf{J} \times \mathbf{B} \quad \text { where } \quad M_{i}=\frac{\partial T_{i j}}{\partial x_{j}}$$

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• Paper 3, Section II, C

  #Vector Calculus

  2013

  (a) Prove that

  $$\nabla \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\nabla \cdot \mathbf{G})-\mathbf{G}(\nabla \cdot \mathbf{F})+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\mathbf{F} \cdot \nabla) \mathbf{G}$$

  (b) State the divergence theorem for a vector field $\mathbf{F}$ in a closed region $\Omega \subset \mathbb{R}^{3}$ bounded by $\partial \Omega$.

  For a smooth vector field $\mathbf{F}$ and a smooth scalar function $g$ prove that

  $$\int_{\Omega} \mathbf{F} \cdot \nabla g+g \nabla \cdot \mathbf{F} d V=\int_{\partial \Omega} g \mathbf{F} \cdot \mathbf{n} d S,$$

  where $\mathbf{n}$ is the outward unit normal on the surface $\partial \Omega$.

  Use this identity to prove that the solution $u$ to the Laplace equation $\nabla^{2} u=0$ in $\Omega$ with $u=f$ on $\partial \Omega$ is unique, provided it exists.

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• Paper 1, Section I, $1 \mathrm{C}$

  #Vectors and Matrices

  2013

  (a) State de Moivre's theorem and use it to derive a formula for the roots of order $n$ of a complex number $z=a+i b$. Using this formula compute the cube roots of $z=-8$.

  (b) Consider the equation $|z+3 i|=3|z|$ for $z \in \mathbb{C}$. Give a geometric description of the set $S$ of solutions and sketch $S$ as a subset of the complex plane.

  comment

• Paper 1, Section I, A

  #Vectors and Matrices

  2013

  Let $A$ be a real $3 \times 3$ matrix.

  (i) For $B=R_{1} A$ with

  $$R_{1}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{1} & -\sin \theta_{1} \\ 0 & \sin \theta_{1} & \cos \theta_{1} \end{array}\right)$$

  find an angle $\theta_{1}$ so that the element $b_{31}=0$, where $b_{i j}$ denotes the $i j^{\text {th }}$ entry of the matrix $B$.

  (ii) For $C=R_{2} B$ with $b_{31}=0$ and

  $$R_{2}=\left(\begin{array}{ccc} \cos \theta_{2} & -\sin \theta_{2} & 0 \\ \sin \theta_{2} & \cos \theta_{2} & 0 \\ 0 & 0 & 1 \end{array}\right)$$

  show that $c_{31}=0$ and find an angle $\theta_{2}$ so that $c_{21}=0$.

  (iii) For $D=R_{3} C$ with $c_{31}=c_{21}=0$ and

  $$R_{3}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{3} & -\sin \theta_{3} \\ 0 & \sin \theta_{3} & \cos \theta_{3} \end{array}\right)$$

  show that $d_{31}=d_{21}=0$ and find an angle $\theta_{3}$ so that $d_{32}=0$.

  (iv) Deduce that any real $3 \times 3$ matrix can be written as a product of an orthogonal matrix and an upper triangular matrix.

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• Paper 1, Section II, $5 \mathrm{C}$

  #Vectors and Matrices

  2013

  Let $\mathbf{x}$ and $\mathbf{y}$ be non-zero vectors in $\mathbb{R}^{n}$. What is meant by saying that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent? What is the dimension of the subspace of $\mathbb{R}^{n}$ spanned by $\mathbf{x}$ and $\mathbf{y}$ if they are (1) linearly independent, (2) linearly dependent?

  Define the scalar product $\mathbf{x} \cdot \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$. Define the corresponding norm $\|\mathbf{x}\|$ of $\mathbf{x} \in \mathbb{R}^{n}$. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality. Under what condition does equality hold in the Cauchy-Schwarz inequality?

  Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ be unit vectors in $\mathbb{R}^{3}$. Let

  $$S=\mathbf{x} \cdot \mathbf{y}+\mathbf{y} \cdot \mathbf{z}+\mathbf{z} \cdot \mathbf{x}$$

  Show that for any fixed, linearly independent vectors $\mathbf{x}$ and $\mathbf{y}$, the minimum of $S$ over $\mathbf{z}$ is attained when $\mathbf{z}=\lambda(\mathbf{x}+\mathbf{y})$ for some $\lambda \in \mathbb{R}$, and that for this value of $\lambda$ we have

  (i) $\lambda \leqslant-\frac{1}{2}$ (for any choice of $\mathbf{x}$ and $\left.\mathbf{y}\right)$;

  (ii) $\lambda=-1$ and $S=-\frac{3}{2}$ in the case where $\mathbf{x} \cdot \mathbf{y}=\cos \frac{2 \pi}{3}$.

  comment

• Paper 1, Section II, $6 \mathrm{~A}$

  #Vectors and Matrices

  2013

  Define the kernel and the image of a linear map $\alpha$ from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$.

  Let $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{m}\right\}$ be a basis of $\mathbb{R}^{m}$ and $\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n}\right\}$ a basis of $\mathbb{R}^{n}$. Explain how to represent $\alpha$ by a matrix $A$ relative to the given bases.

  A second set of bases $\left\{\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \ldots, \mathbf{e}_{m}^{\prime}\right\}$ and $\left\{\mathbf{f}_{1}^{\prime}, \mathbf{f}_{2}^{\prime}, \ldots, \mathbf{f}_{n}^{\prime}\right\}$ is now used to represent $\alpha$ by a matrix $A^{\prime}$. Relate the elements of $A^{\prime}$ to the elements of $A$.

  Let $\beta$ be a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ defined by

  $$\beta\left(\begin{array}{l} 1 \\ 1 \end{array}\right)=\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right), \quad \beta\left(\begin{array}{c} 1 \\ -1 \end{array}\right)=\left(\begin{array}{l} 6 \\ 4 \\ 2 \end{array}\right)$$

  Either find one or more $\mathbf{x}$ in $\mathbb{R}^{2}$ such that

  $$\beta \mathbf{x}=\left(\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right)$$

  or explain why one cannot be found.

  Let $\gamma$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ defined by

  $$\gamma\left(\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right)=\left(\begin{array}{l} 1 \\ 3 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)=\left(\begin{array}{c} -2 \\ 1 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

  Find the kernel of $\gamma$.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2013

  (a) Let $\lambda_{1}, \ldots, \lambda_{d}$ be distinct eigenvalues of an $n \times n$ matrix $A$, with corresponding eigenvectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}$. Prove that the set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}\right\}$ is linearly independent.

  (b) Consider the quadric surface $Q$ in $\mathbb{R}^{3}$ defined by

  $$2 x^{2}-4 x y+5 y^{2}-z^{2}+6 \sqrt{5} y=0 .$$

  Find the position of the origin $\tilde{O}$ and orthonormal coordinate basis vectors $\tilde{\mathbf{e}}_{1}, \tilde{\mathbf{e}}_{2}$ and $\tilde{\mathbf{e}}_{3}$, for a coordinate system $(\tilde{x}, \tilde{y}, \tilde{z})$ in which $Q$ takes the form

  $$\alpha \tilde{x}^{2}+\beta \tilde{y}^{2}+\gamma \tilde{z}^{2}=1 .$$

  Also determine the values of $\alpha, \beta$ and $\gamma$, and describe the surface geometrically.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2013

  (a) Let $A$ and $A^{\prime}$ be the matrices of a linear map $L$ on $\mathbb{C}^{2}$ relative to bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ respectively. In this question you may assume without proof that $A$ and $A^{\prime}$ are similar.

  (i) State how the matrix $A$ of $L$ relative to the basis $\mathcal{B}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is constructed from $L$ and $\mathcal{B}$. Also state how $A$ may be used to compute $L \mathbf{v}$ for any $\mathbf{v} \in \mathbb{C}^{2}$.

  (ii) Show that $A$ and $A^{\prime}$ have the same characteristic equation.

  (iii) Show that for any $k \neq 0$ the matrices

  $$\left(\begin{array}{ll} a & c \\ b & d \end{array}\right) \text { and }\left(\begin{array}{cc} a & c / k \\ b k & d \end{array}\right)$$

  are similar. [Hint: if $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}$ is a basis then so is $\left\{k \mathbf{e}_{1}, \mathbf{e}_{2}\right\}$.]

  (b) Using the results of (a), or otherwise, prove that any $2 \times 2$ complex matrix $M$ with equal eigenvalues is similar to one of

  $$\left(\begin{array}{ll} a & 0 \\ 0 & a \end{array}\right) \text { and }\left(\begin{array}{ll} a & 1 \\ 0 & a \end{array}\right) \text { with } a \in \mathbb{C} .$$

  (c) Consider the matrix

  $$B(r)=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)$$

  Show that there is a real value $r_{0}>0$ such that $B\left(r_{0}\right)$ is an orthogonal matrix. Show that $B\left(r_{0}\right)$ is a rotation and find the axis and angle of the rotation.

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• Paper 3, Section I, $2 F$

  #Analysis II

  2013

  For each of the following sequences of functions on $[0,1]$, indexed by $n=1,2, \ldots$, determine whether or not the sequence has a pointwise limit, and if so, determine whether or not the convergence to the pointwise limit is uniform.

  1. $f_{n}(x)=1 /\left(1+n^{2} x^{2}\right)$

  2. $g_{n}(x)=n x(1-x)^{n}$

  3. $h_{n}(x)=\sqrt{n} x(1-x)^{n}$

  comment

• Paper 4, Section I, $3 F$

  #Analysis II

  2013

  State and prove the chain rule for differentiable mappings $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $G: \mathbb{R}^{m} \rightarrow \mathbb{R}^{k}$.

  Suppose now $F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ has image lying on the unit circle in $\mathbb{R}^{2}$. Prove that the determinant $\operatorname{det}\left(\left.D F\right|_{x}\right)$ vanishes for every $x \in \mathbb{R}^{2}$.

  comment

• Paper 2, Section I, F

  #Analysis II

  2013

  Let $\mathcal{C}[a, b]$ denote the vector space of continuous real-valued functions on the interval $[a, b]$, and let $\mathcal{C}^{\prime}[a, b]$ denote the subspace of continuously differentiable functions.

  Show that $\|f\|_{1}=\max |f|+\max \left|f^{\prime}\right|$ defines a norm on $\mathcal{C}^{\prime}[a, b]$. Show furthermore that the map $\Phi: f \mapsto f^{\prime}((a+b) / 2)$ takes the closed unit ball $\left\{\|f\|_{1} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$.

  If instead we had used the norm $\|f\|_{0}=\max |f|$ restricted from $\mathcal{C}[a, b]$ to $\mathcal{C}^{\prime}[a, b]$, would $\Phi$ take the closed unit ball $\left\{\|f\|_{0} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$ ? Justify your answer.

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• Paper 1, Section II, F

  #Analysis II

  2013

  Define what it means for a sequence of functions $k_{n}: A \rightarrow \mathbb{R}, n=1,2, \ldots$, to converge uniformly on an interval $A \subset \mathbb{R}$.

  By considering the functions $k_{n}(x)=\frac{\sin (n x)}{\sqrt{n}}$, or otherwise, show that uniform convergence of a sequence of differentiable functions does not imply uniform convergence of their derivatives.

  Now suppose $k_{n}(x)$ is continuously differentiable on $A$ for each $n$, that $k_{n}\left(x_{0}\right)$ converges as $n \rightarrow \infty$ for some $x_{0} \in A$, and moreover that the derivatives $k_{n}^{\prime}(x)$ converge uniformly on $A$. Prove that $k_{n}(x)$ converges to a continuously differentiable function $k(x)$ on $A$, and that

  $$k^{\prime}(x)=\lim _{n \rightarrow \infty} k_{n}^{\prime}(x)$$

  Hence, or otherwise, prove that the function

  $$\sum_{n=1}^{\infty} \frac{x^{n} \sin (n x)}{n^{3}+1}$$

  is continuously differentiable on $(-1,1)$.

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• Paper 4, Section II, F

  #Analysis II

  2013

  State the contraction mapping theorem.

  A metric space $(X, d)$ is bounded if $\{d(x, y) \mid x, y \in X\}$ is a bounded subset of $\mathbb{R}$. Suppose $(X, d)$ is complete and bounded. Let $\operatorname{Maps}(X, X)$ denote the set of continuous $\operatorname{maps}$ from $X$ to itself. For $f, g \in \operatorname{Maps}(X, X)$, let

  $$\delta(f, g)=\sup _{x \in X} d(f(x), g(x))$$

  Prove that $(\operatorname{Maps}(X, X), \delta)$ is a complete metric space. Is the subspace $\mathcal{C} \subset \operatorname{Maps}(X, X)$ of contraction mappings a complete subspace?

  Let $\tau: \mathcal{C} \rightarrow X$ be the map which associates to any contraction its fixed point. Prove that $\tau$ is continuous.

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• Paper 3, Section II, F

  #Analysis II

  2013

  For each of the following statements, provide a proof or justify a counterexample.

  1. The norms $\|x\|_{1}=\sum_{i=1}^{n}\left|x_{i}\right|$ and $\|x\|_{\infty}=\max _{1 \leqslant i \leqslant n}\left|x_{i}\right|$ on $\mathbb{R}^{n}$ are Lipschitz equivalent.

  2. The norms $\|x\|_{1}=\sum_{i=1}^{\infty}\left|x_{i}\right|$ and $\|x\|_{\infty}=\max _{i}\left|x_{i}\right|$ on the vector space of sequences $\left(x_{i}\right)_{i \geqslant 1}$ with $\sum\left|x_{i}\right|<\infty$ are Lipschitz equivalent.

  3. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces, there is some $N$ for which $\|\phi(x)\| \leqslant N$ for every $x \in V$ with $\|x\| \leqslant 1$.

  4. Given a linear function $\phi: V \rightarrow W$ between normed real vector spaces for which there is some $N$ for which $\|\phi(x)\| \leqslant N$ for every $x \in V$ with $\|x\| \leqslant 1$, then $\phi$ is continuous.

  5. The uniform norm $\|f\|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ for which $f(x)=0$ for $|x|$ sufficiently large.

  6. The uniform norm $\|f\|=\sup _{x \in \mathbb{R}}|f(x)|$ is complete on the vector space of continuous real-valued functions $f$ on $\mathbb{R}$ which are bounded.

  comment

• Paper 2, Section II, F

  #Analysis II

  2013

  Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subset \mathbb{R}^{2}$. Suppose that on $U$ the partial derivatives $D_{1} f, D_{2} f, D_{1} D_{2} f$ and $D_{2} D_{1} f$ exist and are continuous. Prove that $D_{1} D_{2} f=D_{2} D_{1} f$ on $U$.

  If $f$ is infinitely differentiable, and $m \in \mathbb{N}$, what is the maximum number of distinct $m$-th order partial derivatives that $f$ may have on $U$ ?

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

  $$f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

  Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

  $$g(x, y)= \begin{cases}\frac{x y\left(x^{4}-y^{4}\right)}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

  For each of $f$ and $g$, determine whether they are (i) differentiable, (ii) infinitely differentiable at the origin. Briefly justify your answers.

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• Paper 4, Section I, E

  #Complex Analysis

  2013

  State Rouché's theorem. How many roots of the polynomial $z^{8}+3 z^{7}+6 z^{2}+1$ are contained in the annulus $1<|z|<2$ ?

  comment

• Paper 1, Section II, E

  #Complex Analysis or Complex Methods

  2013

  Suppose $p(z)$ is a polynomial of even degree, all of whose roots satisfy $|z|<R$. Explain why there is a holomorphic (i.e. analytic) function $h(z)$ defined on the region $R<|z|<\infty$ which satisfies $h(z)^{2}=p(z)$. We write $h(z)=\sqrt{p(z)}$

  By expanding in a Laurent series or otherwise, evaluate

  $$\int_{C} \sqrt{z^{4}-z} d z$$

  where $C$ is the circle of radius 2 with the anticlockwise orientation. (Your answer will be well-defined up to a factor of $\pm 1$, depending on which square root you pick.)

  comment

• Paper 3, Section II, E

  #Complex Analysis

  2013

  Let $D=\{z \in \mathbb{C}|| z \mid<1\}$ be the open unit disk, and let $C$ be its boundary (the unit circle), with the anticlockwise orientation. Suppose $\phi: C \rightarrow \mathbb{C}$ is continuous. Stating clearly any theorems you use, show that

  $$g_{\phi}(w)=\frac{1}{2 \pi i} \int_{C} \frac{\phi(z)}{z-w} d z$$

  is an analytic function of $w$ for $w \in D$.

  Now suppose $\phi$ is the restriction of a holomorphic function $F$ defined on some annulus $1-\epsilon<|z|<1+\epsilon$. Show that $g_{\phi}(w)$ is the restriction of a holomorphic function defined on the open disc $|w|<1+\epsilon$.

  Let $f_{\phi}:[0,2 \pi] \rightarrow \mathbb{C}$ be defined by $f_{\phi}(\theta)=\phi\left(e^{i \theta}\right)$. Express the coefficients in the power series expansion of $g_{\phi}$ centered at 0 in terms of $f_{\phi}$.

  Let $n \in \mathbb{Z}$. What is $g_{\phi}$ in the following cases?

  1. $\phi(z)=z^{n}$.

  2. $\phi(z)=\bar{z}^{n}$.

  3. $\phi(z)=(\operatorname{Re} z)^{2}$.

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• Paper 1, Section I, $2 \mathrm{D}$

  #Complex Analysis or Complex Methods

  2013

  Classify the singularities (in the finite complex plane) of the following functions: (i) $\frac{1}{(\cosh z)^{2}}$; (ii) $\frac{1}{\cos (1 / z)}$; (iii) $\frac{1}{\log z} \quad(-\pi<\arg z<\pi)$; (iv) $\frac{z^{\frac{1}{2}}-1}{\sin \pi z} \quad(-\pi<\arg z<\pi)$.

  comment

• Paper 2, Section II, 13D Let

  #Complex Analysis or Complex Methods

  2013

  $$I=\oint_{C} \frac{e^{i z^{2} / \pi}}{1+e^{-2 z}} d z$$

  where $C$ is the rectangle with vertices at $\pm R$ and $\pm R+i \pi$, traversed anti-clockwise.

  (i) Show that $I=\frac{\pi(1+i)}{\sqrt{2}}$.

  (ii) Assuming that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$, show that

  $$\int_{-\infty}^{\infty} e^{i x^{2} / \pi} d x=\frac{\pi(1+i)}{\sqrt{2}}$$

  (iii) Justify briefly the assumption that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$.

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• Paper 3, Section I, D

  #Complex Methods

  2013

  Let $y(t)=0$ for $t<0$, and let $\lim _{t \rightarrow 0^{+}} y(t)=y_{0}$.

  (i) Find the Laplace transforms of $H(t)$ and $t H(t)$, where $H(t)$ is the Heaviside step function.

  (ii) Given that the Laplace transform of $y(t)$ is $\widehat{y}(s)$, find expressions for the Laplace transforms of $\dot{y}(t)$ and $y(t-1)$.

  (iii) Use Laplace transforms to solve the equation

  $$\dot{y}(t)-y(t-1)=H(t)-(t-1) H(t-1)$$

  in the case $y_{0}=0$.

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• Paper 4, Section II, D

  #Complex Methods

  2013

  Let $C_{1}$ and $C_{2}$ be the circles $x^{2}+y^{2}=1$ and $5 x^{2}-4 x+5 y^{2}=0$, respectively, and let $D$ be the (finite) region between the circles. Use the conformal mapping

  $$w=\frac{z-2}{2 z-1}$$

  to solve the following problem:

  $$\nabla^{2} \phi=0 \text { in } D \text { with } \phi=1 \text { on } C_{1} \text { and } \phi=2 \text { on } C_{2}$$

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• Paper 2, Section I, D

  #Electromagnetism

  2013

  Use Maxwell's equations to obtain the equation of continuity

  $$\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{J}=0$$

  Show that, for a body made from material of uniform conductivity $\sigma$, the charge density at any fixed internal point decays exponentially in time. If the body is finite and isolated, explain how this result can be consistent with overall charge conservation.

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• Paper 4, Section I, D

  #Electromagnetism

  2013

  The infinite plane $z=0$ is earthed and the infinite plane $z=d$ carries a charge of $\sigma$ per unit area. Find the electrostatic potential between the planes.

  Show that the electrostatic energy per unit area (of the planes $z=$ constant) between the planes can be written as either $\frac{1}{2} \sigma^{2} d / \epsilon_{0}$ or $\frac{1}{2} \epsilon_{0} V^{2} / d$, where $V$ is the potential at $z=d$.

  The distance between the planes is now increased by $\alpha d$, where $\alpha$ is small. Show that the change in the energy per unit area is $\frac{1}{2} \sigma V \alpha$ if the upper plane $(z=d)$ is electrically isolated, and is approximately $-\frac{1}{2} \sigma V \alpha$ if instead the potential on the upper plane is maintained at $V$. Explain briefly how this difference can be accounted for.

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• Paper 1, Section II, $16 \mathrm{D}$

  #Electromagnetism

  2013

  Briefly explain the main assumptions leading to Drude's theory of conductivity. Show that these assumptions lead to the following equation for the average drift velocity $\langle\mathbf{v}(t)\rangle$ of the conducting electrons:

  $$\frac{d\langle\mathbf{v}\rangle}{d t}=-\tau^{-1}\langle\mathbf{v}\rangle+(e / m) \mathbf{E}$$

  where $m$ and $e$ are the mass and charge of each conducting electron, $\tau^{-1}$ is the probability that a given electron collides with an ion in unit time, and $\mathbf{E}$ is the applied electric field.

  Given that $\langle\mathbf{v}\rangle=\mathbf{v}_{0} e^{-i \omega t}$ and $\mathbf{E}=\mathbf{E}_{0} e^{-i \omega t}$, where $\mathbf{v}_{0}$ and $\mathbf{E}_{0}$ are independent of $t$, show that

  $$\mathbf{J}=\sigma \mathbf{E}$$

  Here, $\sigma=\sigma_{s} /(1-i \omega \tau), \sigma_{s}=n e^{2} \tau / m$ and $n$ is the number of conducting electrons per unit volume.

  Now let $\mathbf{v}_{0}=\widetilde{\mathbf{v}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$ and $\mathbf{E}_{0}=\widetilde{\mathbf{E}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$, where $\widetilde{\mathbf{v}}_{0}$ and $\widetilde{\mathbf{E}}_{0}$ are constant. Assuming that $(*)$ remains valid, use Maxwell's equations (taking the charge density to be everywhere zero but allowing for a non-zero current density) to show that

  $$k^{2}=\frac{\omega^{2}}{c^{2}} \epsilon_{r}$$

  where the relative permittivity $\epsilon_{r}=1+i \sigma /\left(\omega \epsilon_{0}\right)$ and $k=|\mathbf{k}|$.

  In the case $\omega \tau \gg 1$ and $\omega<\omega_{p}$, where $\omega_{p}^{2}=\sigma_{s} / \tau \epsilon_{0}$, show that the wave decays exponentially with distance inside the conductor.

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• Paper 3, Section II, D

  #Electromagnetism

  2013

  Three sides of a closed rectangular circuit $C$ are fixed and one is moving. The circuit lies in the plane $z=0$ and the sides are $x=0, y=0, x=a(t), y=b$, where $a(t)$ is a given function of time. A magnetic field $\mathbf{B}=\left(0,0, \frac{\partial f}{\partial x}\right)$ is applied, where $f(x, t)$ is a given function of $x$ and $t$ only. Find the magnetic flux $\Phi$ of $\mathbf{B}$ through the surface $S$ bounded by $C$.

  Find an electric field $\mathbf{E}_{\mathbf{0}}$ that satisfies the Maxwell equation

  $$\boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

  and then write down the most general solution $\mathbf{E}$ in terms of $\mathbf{E}_{0}$ and an undetermined scalar function independent of $f$.

  Verify that

  $$\oint_{C}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t},$$

  where $\mathbf{v}$ is the velocity of the relevant side of $C$. Interpret the left hand side of this equation.

  If a unit current flows round $C$, what is the rate of work required to maintain the motion of the moving side of the rectangle? You should ignore any electromagnetic fields produced by the current.

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• Paper 2, Section II, D

  #Electromagnetism

  2013

  Starting with the expression

  $$\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) d V^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$$

  for the magnetic vector potential at the point $r$ due to a current distribution of density $\mathbf{J}(\mathbf{r})$, obtain the Biot-Savart law for the magnetic field due to a current $I$ flowing in a simple loop $C$ :

  $$\mathbf{B}(\mathbf{r})=-\frac{\mu_{0} I}{4 \pi} \oint_{C} \frac{d \mathbf{r}^{\prime} \times\left(\mathbf{r}^{\prime}-\mathbf{r}\right)}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|^{3}} \quad(\mathbf{r} \notin C) .$$

  Verify by direct differentiation that this satisfies $\boldsymbol{\nabla} \times \mathbf{B}=\mathbf{0}$. You may use without proof the identity $\boldsymbol{\nabla} \times(\mathbf{a} \times \mathbf{v})=\mathbf{a}(\boldsymbol{\nabla} \cdot \mathbf{v})-(\mathbf{a} \cdot \boldsymbol{\nabla}) \mathbf{v}$, where $\mathbf{a}$ is a constant vector and $\mathbf{v}$ is a vector field.

  Given that $C$ is planar, and is described in cylindrical polar coordinates by $z=0$, $r=f(\theta)$, show that the magnetic field at the origin is

  $$\widehat{\mathbf{z}} \frac{\mu_{0} I}{4 \pi} \oint \frac{d \theta}{f(\theta)}$$

  If $C$ is the ellipse $r(1-e \cos \theta)=\ell$, find the magnetic field at the focus due to a current $I$.

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• Paper 1, Section I, A

  #Fluid Dynamics

  2013

  A two-dimensional flow is given by

  $$\mathbf{u}=(x,-y+t)$$

  Show that the flow is both irrotational and incompressible. Find a stream function $\psi(x, y)$ such that $\mathbf{u}=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$. Sketch the streamlines at $t=0$.

  Find the pathline of a fluid particle that passes through $\left(x_{0}, y_{0}\right)$ at $t=0$ in the form $y=f\left(x, x_{0}, y_{0}\right)$ and sketch the pathline for $x_{0}=1, y_{0}=1 .$

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• Paper 2, Section I, A

  #Fluid Dynamics

  2013

  An incompressible, inviscid fluid occupies the region beneath the free surface $y=\eta(x, t)$ and moves with a velocity field determined by the velocity potential $\phi(x, y, t) .$ Gravity acts in the $-y$ direction. You may assume Bernoulli's integral of the equation of motion:

  $$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}+g y=F(t)$$

  Give the kinematic and dynamic boundary conditions that must be satisfied by $\phi$ on $y=\eta(x, t)$.

  In the absence of waves, the fluid has constant uniform velocity $U$ in the $x$ direction. Derive the linearised form of the boundary conditions for small amplitude waves.

  Assume that the free surface and velocity potential are of the form:

  $$\begin{aligned} \eta &=a e^{i(k x-\omega t)} \\ \phi &=U x+i b e^{k y} e^{i(k x-\omega t)} \end{aligned}$$

  (where implicitly the real parts are taken). Show that

  $$(\omega-k U)^{2}=g k$$

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• Paper 1, Section II, A

  #Fluid Dynamics

  2013

  Starting from the Euler momentum equation, derive the form of Bernoulli's equation appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.

  Water of density $\rho$ is driven through a horizontal tube of length $L$ and internal radius $a$ from a water-filled balloon attached to one end of the tube. Assume that the pressure exerted by the balloon is proportional to its current volume (in excess of atmospheric pressure). Also assume that water exits the tube at atmospheric pressure, and that gravity may be neglected. Show that the time for the balloon to empty does not depend on its initial volume. Find the maximum speed of water exiting the pipe.

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• Paper 4, Section II, A

  #Fluid Dynamics

  2013

  The axisymmetric, irrotational flow generated by a solid sphere of radius $a$ translating at velocity $U$ in an inviscid, incompressible fluid is represented by a velocity potential $\phi(r, \theta)$. Assume the fluid is at rest far away from the sphere. Explain briefly why $\nabla^{2} \phi=0$.

  By trying a solution of the form $\phi(r, \theta)=f(r) g(\theta)$, show that

  $$\phi=-\frac{U a^{3} \cos \theta}{2 r^{2}}$$

  and write down the fluid velocity.

  Show that the total kinetic energy of the fluid is $k M U^{2} / 4$ where $M$ is the mass of the sphere and $k$ is the ratio of the density of the fluid to the density of the sphere.

  A heavy sphere (i.e. $k<1$ ) is released from rest in an inviscid fluid. Determine its speed after it has fallen a distance $h$ in terms of $M, k, g$ and $h$.

  Note, in spherical polars:

  $$\begin{gathered} \boldsymbol{\nabla} \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{\mathbf{r}}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta} \\ \nabla^{2} \phi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi}{\partial \theta}\right) \end{gathered}$$

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• Paper 3, Section II, A

  #Fluid Dynamics

  2013

  A layer of incompressible fluid of density $\rho$ and viscosity $\mu$ flows steadily down a plane inclined at an angle $\theta$ to the horizontal. The layer is of uniform thickness $h$ measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using coordinates $x$ parallel to the plane (in steepest downwards direction) and $y$ normal to the plane, write down the equations of motion and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields and show that the volume flux down the plane is

  $$\frac{\rho g h^{3} \sin \theta}{3 \mu}$$

  Consider now the case where a second layer of fluid, of uniform thickness $\alpha h$, viscosity $\beta \mu$ and density $\rho$, flows steadily on top of the first layer. Explain why one of the appropriate boundary conditions between the two fluids is

  $$\mu \frac{\partial}{\partial y} u\left(h_{-}\right)=\beta \mu \frac{\partial}{\partial y} u\left(h_{+}\right),$$

  where $u$ is the component of velocity in the $x$ direction and $h_{-}$and $h_{+}$refer to just below and just above the boundary respectively. Determine the velocity field in each layer.

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• Paper 1, Section I, F

  #Geometry

  2013

  Let $l_{1}$ and $l_{2}$ be ultraparallel geodesics in the hyperbolic plane. Prove that the $l_{i}$ have a unique common perpendicular.

  Suppose now $l_{1}, l_{2}, l_{3}$ are pairwise ultraparallel geodesics in the hyperbolic plane. Can the three common perpendiculars be pairwise disjoint? Must they be pairwise disjoint? Briefly justify your answers.

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• Paper 3, Section I, F

  #Geometry

  2013

  Let $S$ be a surface with Riemannian metric having first fundamental form $d u^{2}+G(u, v) d v^{2}$. State a formula for the Gauss curvature $K$ of $S$.

  Suppose that $S$ is flat, so $K$ vanishes identically, and that $u=0$ is a geodesic on $S$ when parametrised by arc-length. Using the geodesic equations, or otherwise, prove that $G(u, v) \equiv 1$, i.e. $S$ is locally isometric to a plane.

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• Paper 2, Section II, F

  #Geometry

  2013

  Let $A$ and $B$ be disjoint circles in $\mathbb{C}$. Prove that there is a Möbius transformation which takes $A$ and $B$ to two concentric circles.

  A collection of circles $X_{i} \subset \mathbb{C}, 0 \leqslant i \leqslant n-1$, for which

  1. $X_{i}$ is tangent to $A, B$ and $X_{i+1}$, where indices are $\bmod n$;

  2. the circles are disjoint away from tangency points;

  is called a constellation on $(A, B)$. Prove that for any $n \geqslant 2$ there is some pair $(A, B)$ and a constellation on $(A, B)$ made up of precisely $n$ circles. Draw a picture illustrating your answer.

  Given a constellation on $(A, B)$, prove that the tangency points $X_{i} \cap X_{i+1}$ for $0 \leqslant i \leqslant n-1$ all lie on a circle. Moreover, prove that if we take any other circle $Y_{0}$ tangent to $A$ and $B$, and then construct $Y_{i}$ for $i \geqslant 1$ inductively so that $Y_{i}$ is tangent to $A, B$ and $Y_{i-1}$, then we will have $Y_{n}=Y_{0}$, i.e. the chain of circles will again close up to form a constellation.

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• Paper 3, Section II, F

  #Geometry

  2013

  Show that the set of all straight lines in $\mathbb{R}^{2}$ admits the structure of an abstract smooth surface $S$. Show that $S$ is an open Möbius band (i.e. the Möbius band without its boundary circle), and deduce that $S$ admits a Riemannian metric with vanishing Gauss curvature.

  Show that there is no metric $d: S \times S \rightarrow \mathbb{R}_{\geqslant 0}$, in the sense of metric spaces, which

  1. induces the locally Euclidean topology on $S$ constructed above;

  2. is invariant under the natural action on $S$ of the group of translations of $\mathbb{R}^{2}$.

  Show that the set of great circles on the two-dimensional sphere admits the structure of a smooth surface $S^{\prime}$. Is $S^{\prime}$ homeomorphic to $S$ ? Does $S^{\prime}$ admit a Riemannian metric with vanishing Gauss curvature? Briefly justify your answers.

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• Paper 4, Section II, F

  #Geometry

  2013

  Let $\eta$ be a smooth curve in the $x z$-plane $\eta(s)=(f(s), 0, g(s))$, with $f(s)>0$ for every $s \in \mathbb{R}$ and $f^{\prime}(s)^{2}+g^{\prime}(s)^{2}=1$. Let $S$ be the surface obtained by rotating $\eta$ around the $z$-axis. Find the first fundamental form of $S$.

  State the equations for a curve $\gamma:(a, b) \rightarrow S$ parametrised by arc-length to be a geodesic.

  A parallel on $S$ is the closed circle swept out by rotating a single point of $\eta$. Prove that for every $n \in \mathbb{Z}_{>0}$ there is some $\eta$ for which exactly $n$ parallels are geodesics. Sketch possible such surfaces $S$ in the cases $n=1$ and $n=2$.

  If every parallel is a geodesic, what can you deduce about $S$ ? Briefly justify your answer.

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• Paper 3, Section I, $1 G$

  #Groups, Rings and Modules

  2013

  Define the notion of a free module over a ring. When $R$ is a PID, show that every ideal of $R$ is free as an $R$-module.

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• Paper 4, Section I, $2 G$

  #Groups, Rings and Modules

  2013

  Let $p$ be a prime number, and $G$ be a non-trivial finite group whose order is a power of $p$. Show that the size of every conjugacy class in $G$ is a power of $p$. Deduce that the centre $Z$ of $G$ has order at least $p$.

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• Paper 2, Section I, G

  #Groups, Rings and Modules

  2013

  Show that every Euclidean domain is a PID. Define the notion of a Noetherian ring, and show that $\mathbb{Z}[i]$ is Noetherian by using the fact that it is a Euclidean domain.

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• Paper 1, Section II, G

  #Groups, Rings and Modules

  2013

  (i) Consider the group $G=G L_{2}(\mathbb{R})$ of all 2 by 2 matrices with entries in $\mathbb{R}$ and non-zero determinant. Let $T$ be its subgroup consisting of all diagonal matrices, and $N$ be the normaliser of $T$ in $G$. Show that $N$ is generated by $T$ and $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, and determine the quotient group $N / T$.

  (ii) Now let $p$ be a prime number, and $F$ be the field of integers modulo $p$. Consider the group $G=G L_{2}(F)$ as above but with entries in $F$, and define $T$ and $N$ similarly. Find the order of the group $N$.

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• Paper 4, Section II, 11G

  #Groups, Rings and Modules

  2013

  Let $R$ be an integral domain, and $M$ be a finitely generated $R$-module.

  (i) Let $S$ be a finite subset of $M$ which generates $M$ as an $R$-module. Let $T$ be a maximal linearly independent subset of $S$, and let $N$ be the $R$-submodule of $M$ generated by $T$. Show that there exists a non-zero $r \in R$ such that $r x \in N$ for every $x \in M$.

  (ii) Now assume $M$ is torsion-free, i.e. $r x=0$ for $r \in R$ and $x \in M$ implies $r=0$ or $x=0$. By considering the map $M \rightarrow N$ mapping $x$ to $r x$ for $r$ as in (i), show that every torsion-free finitely generated $R$-module is isomorphic to an $R$-submodule of a finitely generated free $R$-module.

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• Paper 3, Section II, G

  #Groups, Rings and Modules

  2013

  Let $R=\mathbb{C}[X, Y]$ be the polynomial ring in two variables over the complex numbers, and consider the principal ideal $I=\left(X^{3}-Y^{2}\right)$ of $R$.

  (i) Using the fact that $R$ is a UFD, show that $I$ is a prime ideal of $R$. [Hint: Elements in $\mathbb{C}[X, Y]$ are polynomials in $Y$ with coefficients in $\mathbb{C}[X] .]$

  (ii) Show that $I$ is not a maximal ideal of $R$, and that it is contained in infinitely many distinct proper ideals in $R$.

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• Paper 2, Section II, G

  #Groups, Rings and Modules

  2013

  (i) State the structure theorem for finitely generated modules over Euclidean domains.

  (ii) Let $\mathbb{C}[X]$ be the polynomial ring over the complex numbers. Let $M$ be a $\mathbb{C}[X]$ module which is 4-dimensional as a $\mathbb{C}$-vector space and such that $(X-2)^{4} \cdot x=0$ for all $x \in M$. Find all possible forms we obtain when we write $M \cong \bigoplus_{i=1}^{m} \mathbb{C}[X] /\left(P_{i}^{n_{i}}\right)$ for irreducible $P_{i} \in \mathbb{C}[X]$ and $n_{i} \geqslant 1$.

  (iii) Consider the quotient ring $M=\mathbb{C}[X] /\left(X^{3}+X\right)$ as a $\mathbb{C}[X]$-module. Show that $M$ is isomorphic as a $\mathbb{C}[X]$-module to the direct sum of three copies of $\mathbb{C}$. Give the isomorphism and its inverse explicitly.

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• Paper 4, Section I, E

  #Linear Algebra

  2013

  What is a quadratic form on a finite dimensional real vector space $V$ ? What does it mean for two quadratic forms to be isomorphic (i.e. congruent)? State Sylvester's law of inertia and explain the definition of the quantities which appear in it. Find the signature of the quadratic form on $\mathbb{R}^{3}$ given by $q(\mathbf{v})=\mathbf{v}^{T} A \mathbf{v}$, where

  $$A=\left(\begin{array}{ccc} -2 & 1 & 6 \\ 1 & -1 & -3 \\ 6 & -3 & 1 \end{array}\right)$$

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• Paper 2, Section I, E

  #Linear Algebra

  2013

  If $A$ is an $n \times n$ invertible Hermitian matrix, let

  $$U_{A}=\left\{U \in M_{n \times n}(\mathbb{C}) \mid \bar{U}^{T} A U=A\right\}$$

  Show that $U_{A}$ with the operation of matrix multiplication is a group, and that det $U$ has norm 1 for any $U \in U_{A}$. What is the relation between $U_{A}$ and the complex Hermitian form defined by $A$ ?

  If $A=I_{n}$ is the $n \times n$ identity matrix, show that any element of $U_{A}$ is diagonalizable.

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• Paper 1, Section I, E

  #Linear Algebra

  2013

  What is the adjugate of an $n \times n$ matrix $A$ ? How is it related to $A^{-1}$ ? Suppose all the entries of $A$ are integers. Show that all the entries of $A^{-1}$ are integers if and only if $\operatorname{det} A=\pm 1$.

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• Paper 1, Section II, E

  #Linear Algebra

  2013

  If $V_{1}$ and $V_{2}$ are vector spaces, what is meant by $V_{1} \oplus V_{2}$ ? If $V_{1}$ and $V_{2}$ are subspaces of a vector space $V$, what is meant by $V_{1}+V_{2}$ ?

  Stating clearly any theorems you use, show that if $V_{1}$ and $V_{2}$ are subspaces of a finite dimensional vector space $V$, then

  $$\operatorname{dim} V_{1}+\operatorname{dim} V_{2}=\operatorname{dim}\left(V_{1} \cap V_{2}\right)+\operatorname{dim}\left(V_{1}+V_{2}\right)$$

  Let $V_{1}, V_{2} \subset \mathbb{R}^{4}$ be subspaces with bases

  $$\begin{gathered} V_{1}=\langle(3,2,4,-1),(1,2,1,-2),(-2,3,3,2)\rangle \\ V_{2}=\langle(1,4,2,4),(-1,1,-1,-1),(3,1,2,0)\rangle . \end{gathered}$$

  Find a basis $\left\langle\mathbf{v}_{1}, \mathbf{v}_{2}\right\rangle$ for $V_{1} \cap V_{2}$ such that the first component of $\mathbf{v}_{1}$ and the second component of $\mathbf{v}_{2}$ are both 0 .

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• Paper 4, Section II, E

  #Linear Algebra

  2013

  What does it mean for an $n \times n$ matrix to be in Jordan form? Show that if $A \in M_{n \times n}(\mathbb{C})$ is in Jordan form, there is a sequence $\left(A_{m}\right)$ of diagonalizable $n \times n$ matrices which converges to $A$, in the sense that the $(i j)$ th component of $A_{m}$ converges to the $(i j)$ th component of $A$ for all $i$ and $j$. [Hint: A matrix with distinct eigenvalues is diagonalizable.] Deduce that the same statement holds for all $A \in M_{n \times n}(\mathbb{C})$.

  Let $V=M_{2 \times 2}(\mathbb{C})$. Given $A \in V$, define a linear map $T_{A}: V \rightarrow V$ by $T_{A}(B)=A B+B A$. Express the characteristic polynomial of $T_{A}$ in terms of the trace and determinant of $A$. [Hint: First consider the case where $A$ is diagonalizable.]

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• Paper 3, Section II, E

  #Linear Algebra

  2013

  Let $V$ and $W$ be finite dimensional real vector spaces and let $T: V \rightarrow W$ be a linear map. Define the dual space $V^{*}$ and the dual map $T^{*}$. Show that there is an isomorphism $\iota: V \rightarrow\left(V^{*}\right)^{*}$ which is canonical, in the sense that $\iota \circ S=\left(S^{*}\right)^{*} \circ \iota$ for any automorphism $S$ of $V$.

  Now let $W$ be an inner product space. Use the inner product to show that there is an injective map from im $T$ to $\operatorname{im} T^{*}$. Deduce that the row rank of a matrix is equal to its column rank.

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• Paper 2, Section II, E

  #Linear Algebra

  2013

  Define what it means for a set of vectors in a vector space $V$ to be linearly dependent. Prove from the definition that any set of $n+1$ vectors in $\mathbb{R}^{n}$ is linearly dependent.

  Using this or otherwise, prove that if $V$ has a finite basis consisting of $n$ elements, then any basis of $V$ has exactly $n$ elements.

  Let $V$ be the vector space of bounded continuous functions on $\mathbb{R}$. Show that $V$ is infinite dimensional.

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• Paper 4, Section I, H

  #Markov Chains

  2013

  Suppose $P$ is the transition matrix of an irreducible recurrent Markov chain with state space $I$. Show that if $x$ is an invariant measure and $x_{k}>0$ for some $k \in I$, then $x_{j}>0$ for all $j \in I$.

  Let

  $$\gamma_{j}^{k}=p_{k j}+\sum_{t=1}^{\infty} \sum_{i_{1} \neq k, \ldots, i_{t} \neq k} p_{k i_{t}} p_{i_{t} i_{t-1}} \cdots p_{i_{1} j}$$

  Give a meaning to $\gamma_{j}^{k}$ and explain why $\gamma_{k}^{k}=1$.

  Suppose $x$ is an invariant measure with $x_{k}=1$. Prove that $x_{j} \geqslant \gamma_{j}^{k}$ for all $j$.

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• Paper 3, Section I, H

  #Markov Chains

  2013

  Prove that if a distribution $\pi$ is in detailed balance with a transition matrix $P$ then it is an invariant distribution for $P$.

  Consider the following model with 2 urns. At each time, $t=0,1, \ldots$ one of the following happens:

  • with probability $\beta$ a ball is chosen at random and moved to the other urn (but nothing happens if both urns are empty);

  • with probability $\gamma$ a ball is chosen at random and removed (but nothing happens if both urns are empty);

  • with probability $\alpha$ a new ball is added to a randomly chosen urn,

  where $\alpha+\beta+\gamma=1$ and $\alpha<\gamma$. State $(i, j)$ denotes that urns 1,2 contain $i$ and $j$ balls respectively. Prove that there is an invariant measure

  $$\lambda_{i, j}=\frac{(i+j) !}{i ! j !}(\alpha / 2 \gamma)^{i+j}$$

  Find the proportion of time for which there are $n$ balls in the system.

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• Paper 1, Section II, 20H

  #Markov Chains

  2013

  A Markov chain has state space $\{a, b, c\}$ and transition matrix

  $$P=\left(\begin{array}{ccc} 0 & 3 / 5 & 2 / 5 \\ 3 / 4 & 0 & 1 / 4 \\ 2 / 3 & 1 / 3 & 0 \end{array}\right)$$

  where the rows $1,2,3$ correspond to $a, b, c$, respectively. Show that this Markov chain is equivalent to a random walk on some graph with 6 edges.

  Let $k(i, j)$ denote the mean first passage time from $i$ to $j$.

  (i) Find $k(a, a)$ and $k(a, b)$.

  (ii) Given $X_{0}=a$, find the expected number of steps until the walk first completes a step from $b$ to $c$.

  (iii) Suppose the distribution of $X_{0}$ is $\left(\pi_{1}, \pi_{2}, \pi_{3}\right)=(5,4,3) / 12$. Let $\tau(a, b)$ be the least $m$ such that $\{a, b\}$ appears as a subsequence of $\left\{X_{0}, X_{1}, \ldots, X_{m}\right\}$. By comparing the distributions of $\left\{X_{0}, X_{1}, \ldots, X_{m}\right\}$ and $\left\{X_{m}, \ldots, X_{1}, X_{0}\right\}$ show that $E \tau(a, b)=E \tau(b, a)$ and that

  $$k(b, a)-k(a, b)=\sum_{i \in\{a, b, c\}} \pi_{i}[k(i, a)-k(i, b)]$$

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• Paper 2, Section II, H

  #Markov Chains

  2013

  (i) Suppose $\left(X_{n}\right)_{n \geqslant 0}$ is an irreducible Markov chain and $f_{i j}=P\left(X_{n}=j\right.$ for some $\left.n \geqslant 1 \mid X_{0}=i\right)$. Prove that $f_{i i} \geqslant f_{i j} f_{j i}$ and that

  $$\sum_{n=0}^{\infty} P_{i}\left(X_{n}=i\right)=\sum_{n=1}^{\infty} f_{i i}^{n-1}$$

  (ii) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a symmetric random walk on the $\mathbb{Z}^{2}$ lattice. Prove that $\left(X_{n}\right)_{n \geqslant 0}$ is recurrent. You may assume, for $n \geqslant 1$,

  $$1 / 2<2^{-2 n} \sqrt{n}\left(\begin{array}{c} 2 n \\ n \end{array}\right)<1$$

  (iii) A princess and monster perform independent random walks on the $\mathbb{Z}^{2}$ lattice. The trajectory of the princess is the symmetric random walk $\left(X_{n}\right)_{n \geqslant 0}$. The monster's trajectory, denoted $\left(Z_{n}\right)_{n \geqslant 0}$, is a sleepy version of an independent symmetric random walk $\left(Y_{n}\right)_{n \geqslant 0}$. Specifically, given an infinite sequence of integers $0=n_{0}<n_{1}<\cdots$, the monster sleeps between these times, so $Z_{n_{i}+1}=\cdots=Z_{n_{i+1}}=Y_{i+1}$. Initially, $X_{0}=(100,0)$ and $Z_{0}=Y_{0}=(0,100)$. The princess is captured if and only if at some future time she and the monster are simultaneously at $(0,0)$.

  Compare the capture probabilities for an active monster, who takes $n_{i+1}=n_{i}+1$ for all $i$, and a sleepy monster, who takes $n_{i}$ spaced sufficiently widely so that

  $$P\left(X_{k}=(0,0) \text { for some } k \in\left\{n_{i}+1, \ldots, n_{i+1}\right\}\right)>1 / 2$$

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• Paper 2, Section I, B

  #Methods

  2013

  Consider the equation

  $$x u_{x}+(x+y) u_{y}=1$$

  subject to the Cauchy data $u(1, y)=y$. Using the method of characteristics, obtain a solution to this equation.

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• Paper 4, Section I, C

  #Methods

  2013

  Show that the general solution of the wave equation

  $$\frac{1}{c^{2}} \frac{\partial^{2} y}{\partial t^{2}}-\frac{\partial^{2} y}{\partial x^{2}}=0$$

  can be written in the form

  $$y(x, t)=f(c t-x)+g(c t+x) .$$

  For the boundary conditions

  $$y(0, t)=y(L, t)=0, \quad t>0,$$

  find the relation between $f$ and $g$ and show that they are $2 L$-periodic. Hence show that

  $$E(t)=\frac{1}{2} \int_{0}^{L}\left(\frac{1}{c^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right) d x$$

  is independent of $t$.

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• Paper 3, Section I, C

  #Methods

  2013

  The solution to the Dirichlet problem on the half-space $D=\{\mathbf{x}=(x, y, z): z>0\}$ :

  $$\nabla^{2} u(\mathbf{x})=0, \quad \mathbf{x} \in D, \quad u(\mathbf{x}) \rightarrow 0 \quad \text { as } \quad|\mathbf{x}| \rightarrow \infty, \quad u(x, y, 0)=h(x, y)$$

  is given by the formula

  $$u\left(\mathbf{x}_{0}\right)=u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x, y) \frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right) d x d y$$

  where $n$ is the outward normal to $\partial D$.

  State the boundary conditions on $G$ and explain how $G$ is related to $G_{3}$, where

  $$G_{3}\left(\mathbf{x}, \mathbf{x}_{0}\right)=-\frac{1}{4 \pi} \frac{1}{\left|\mathbf{x}-\mathbf{x}_{0}\right|}$$

  is the fundamental solution to the Laplace equation in three dimensions.

  Using the method of images find an explicit expression for the function $\frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right)$ in the formula.

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• Paper 1, Section II, B

  #Methods

  2013

  (i) Let $f(x)=x, 0<x \leqslant \pi$. Obtain the Fourier sine series and sketch the odd and even periodic extensions of $f(x)$ over the interval $-2 \pi \leqslant x \leqslant 2 \pi$. Deduce that

  $$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$

  (ii) Consider the eigenvalue problem

  $$\mathcal{L} y=-\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}=\lambda y, \quad \lambda \in \mathbb{R}$$

  with boundary conditions $y(0)=y(\pi)=0$. Find the eigenvalues and corresponding eigenfunctions. Recast $\mathcal{L}$ in Sturm-Liouville form and give the orthogonality condition for the eigenfunctions. Using the Fourier sine series obtained in part (i), or otherwise, and assuming completeness of the eigenfunctions, find a series for $y$ that satisfies

  $$\mathcal{L} y=x e^{-x}$$

  for the given boundary conditions.

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• Paper 3, Section II, C

  #Methods

  2013

  The Laplace equation in plane polar coordinates has the form

  $$\nabla^{2} \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0 .$$

  Using separation of variables, derive the general solution to the equation that is singlevalued in the domain $1<r<2$.

  For

  $$f(\theta)=\sum_{n=1}^{\infty} A_{n} \sin n \theta$$

  solve the Laplace equation in the annulus with the boundary conditions:

  $$\nabla^{2} \phi=0, \quad 1<r<2, \quad \phi(r, \theta)= \begin{cases}f(\theta), & r=1 \\ f(\theta)+1, & r=2\end{cases}$$

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• Paper 2, Section II, B

  #Methods

  2013

  The steady-state temperature distribution $u(x)$ in a uniform rod of finite length satisfies the boundary value problem

  $$\begin{gathered} -D \frac{d^{2}}{d x^{2}} u(x)=f(x), \quad 0<x<l \\ u(0)=0, \quad u(l)=0 \end{gathered}$$

  where $D>0$ is the (constant) diffusion coefficient. Determine the Green's function $G(x, \xi)$ for this problem. Now replace the above homogeneous boundary conditions with the inhomogeneous boundary conditions $u(0)=\alpha, \quad u(l)=\beta$ and give a solution to the new boundary value problem. Hence, obtain the steady-state solution for the following problem with the specified boundary conditions:

  $$\begin{aligned} &-D \frac{\partial^{2}}{\partial x^{2}} u(x, t)+\frac{\partial}{\partial t} u(x, t)=x, \quad 0<x<1 \\ &u(0, t)=1 / D, \quad u(1, t)=2 / D, \quad t>0 \end{aligned}$$

  [You may assume that a steady-state solution exists.]

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• Paper 4, Section II, C

  #Methods

  2013

  Find the inverse Fourier transform $G(x)$ of the function

  $$g(k)=e^{-a|k|}, \quad a>0, \quad-\infty<k<\infty .$$

  Assuming that appropriate Fourier transforms exist, determine the solution $\psi(x, y)$ of

  $$\nabla^{2} \psi=0, \quad-\infty<x<\infty, \quad 0<y<1$$

  with the following boundary conditions

  $$\psi(x, 0)=\delta(x), \quad \psi(x, 1)=\frac{1}{\pi} \frac{1}{x^{2}+1}$$

  Here $\delta(x)$ is the Dirac delta-function.

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• Paper 3, Section I, G

  #Metric and Topological Spaces

  2013

  Let $X$ be a metric space with the metric $d: X \times X \rightarrow \mathbb{R}$.

  (i) Show that if $X$ is compact as a topological space, then $X$ is complete.

  (ii) Show that the completeness of $X$ is not a topological property, i.e. give an example of two metrics $d, d^{\prime}$ on a set $X$, such that the associated topologies are the same, but $(X, d)$ is complete and $\left(X, d^{\prime}\right)$ is not.

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• Paper 2, Section I, G

  #Metric and Topological Spaces

  2013

  Let $X$ be a topological space. Prove or disprove the following statements.

  (i) If $X$ is discrete, then $X$ is compact if and only if it is a finite set.

  (ii) If $Y$ is a subspace of $X$ and $X, Y$ are both compact, then $Y$ is closed in $X$.

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• Paper 1, Section II, G

  #Metric and Topological Spaces

  2013

  Consider the sphere $S^{2}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\}$, a subset of $\mathbb{R}^{3}$, as a subspace of $\mathbb{R}^{3}$ with the Euclidean metric.

  (i) Show that $S^{2}$ is compact and Hausdorff as a topological space.

  (ii) Let $X=S^{2} / \sim$ be the quotient set with respect to the equivalence relation identifying the antipodes, i.e.

  $$(x, y, z) \sim\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \Longleftrightarrow\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(x, y, z) \text { or }(-x,-y,-z)$$

  Show that $X$ is compact and Hausdorff with respect to the quotient topology.

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• Paper 4, Section II, G

  #Metric and Topological Spaces

  2013

  Let $X$ be a topological space. A connected component of $X$ means an equivalence class with respect to the equivalence relation on $X$ defined as:

  $$x \sim y \Longleftrightarrow x, y \text { belong to some connected subspace of } X .$$

  (i) Show that every connected component is a connected and closed subset of $X$.

  (ii) If $X, Y$ are topological spaces and $X \times Y$ is the product space, show that every connected component of $X \times Y$ is a direct product of connected components of $X$ and $Y$.

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• Paper 1, Section I, C

  #Numerical Analysis

  2013

  Determine the nodes $x_{1}, x_{2}$ of the two-point Gaussian quadrature

  $$\int_{0}^{1} f(x) w(x) d x \approx a_{1} f\left(x_{1}\right)+a_{2} f\left(x_{2}\right), \quad w(x)=x$$

  and express the coefficients $a_{1}, a_{2}$ in terms of $x_{1}, x_{2}$. [You don't need to find numerical values of the coefficients.]

  comment

• Paper 4, Section I, C

  #Numerical Analysis

  2013

  For a continuous function $f$, and $k+1$ distinct points $\left\{x_{0}, x_{1}, \ldots, x_{k}\right\}$, define the divided difference $f\left[x_{0}, \ldots, x_{k}\right]$ of order $k$.

  Given $n+1$ points $\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$, let $p_{n} \in \mathbb{P}_{n}$ be the polynomial of degree $n$ that interpolates $f$ at these points. Prove that $p_{n}$ can be written in the Newton form

  $$p_{n}(x)=f\left(x_{0}\right)+\sum_{k=1}^{n} f\left[x_{0}, \ldots, x_{k}\right] \prod_{i=0}^{k-1}\left(x-x_{i}\right)$$

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• Paper 1, Section II, C

  #Numerical Analysis

  2013

  Define the QR factorization of an $m \times n$ matrix $A$ and explain how it can be used to solve the least squares problem of finding the vector $x^{*} \in \mathbb{R}^{n}$ which minimises $\|A x-b\|$, where $b \in \mathbb{R}^{m}, m>n$, and the norm is the Euclidean one.

  Define a Givens rotation $\Omega^{[p, q]}$ and show that it is an orthogonal matrix.

  Using a Givens rotation, solve the least squares problem for

  $$A=\left[\begin{array}{lll} 2 & 1 & 1 \\ 0 & 4 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 0 \end{array}\right], \quad b=\left[\begin{array}{l} 2 \\ 3 \\ 1 \\ 2 \end{array}\right]$$

  giving both $x^{*}$ and $\left\|A x^{*}-b\right\|$.

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• Paper 3, Section II, C

  #Numerical Analysis

  2013

  $$f^{\prime}(0) \approx a_{0} f(0)+a_{1} f(1)+a_{2} f(2)=: \lambda(f)$$

  be a formula of numerical differentiation which is exact on polynomials of degree 2 , and let

  $$e(f)=f^{\prime}(0)-\lambda(f)$$

  be its error.

  Find the values of the coefficients $a_{0}, a_{1}, a_{2}$.

  Using the Peano kernel theorem, find the least constant $c$ such that, for all functions $f \in C^{3}[0,2]$, we have

  $$|e(f)| \leqslant c\left\|f^{\prime \prime \prime}\right\|_{\infty} .$$

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• Paper 2, Section II, C

  #Numerical Analysis

  2013

  Explain briefly what is meant by the convergence of a numerical method for solving the ordinary differential equation

  $$y^{\prime}(t)=f(t, y), \quad t \in[0, T], \quad y(0)=y_{0} .$$

  Prove from first principles that if the function $f$ is sufficiently smooth and satisfies the Lipschitz condition

  $$|f(t, x)-f(t, y)| \leqslant L|x-y|, \quad x, y \in \mathbb{R}, \quad t \in[0, T],$$

  for some $L>0$, then the backward Euler method

  $$y_{n+1}=y_{n}+h f\left(t_{n+1}, y_{n+1}\right)$$

  converges and find the order of convergence.

  Find the linear stability domain of the backward Euler method.

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• Paper 1, Section I, $8 \mathrm{H}$

  #Optimization

  2013

  State sufficient conditions for $p$ and $q$ to be optimal mixed strategies for the row and column players in a zero-sum game with payoff matrix $A$ and value $v$.

  Rowena and Colin play a hide-and-seek game. Rowena hides in one of 3 locations, and then Colin searches them in some order. If he searches in order $i, j, k$ then his search cost is $c_{i}, c_{i}+c_{j}$ or $c_{i}+c_{j}+c_{k}$, depending upon whether Rowena hides in $i, j$ or $k$, respectively, and where $c_{1}, c_{2}, c_{3}$ are all positive. Rowena (Colin) wishes to maximize (minimize) the expected search cost.

  Formulate the payoff matrix for this game.

  Let $c=c_{1}+c_{2}+c_{3}$. Suppose that Colin starts his search in location $i$ with probability $c_{i} / c$, and then, if he does not find Rowena, he searches the remaining two locations in random order. What bound does this strategy place on the value of the game?

  Guess Rowena's optimal hiding strategy, show that it is optimal and find the value of the game.

  comment

• Paper 2, Section I, H

  #Optimization

  2013

  Given a network with a source $A$, a sink $B$, and capacities on directed arcs, define what is meant by a minimum cut.

  The $m$ streets and $n$ intersections of a town are represented by sets of edges $E$ and vertices $V$ of a connected graph. A city planner wishes to make all streets one-way while ensuring it possible to drive away from each intersection along at least $k$ different streets.

  Use a theorem about min-cut and max-flow to prove that the city planner can achieve his goal provided that the following is true:

  $$d(U) \geqslant k|U| \text { for all } U \subseteq V,$$

  where $|U|$ is the size of $U$ and $d(U)$ is the number edges with at least one end in $U$. How could the planner find street directions that achieve his goal?

  [Hint: Consider a network having nodes $A, B$, nodes $a_{1}, \ldots, a_{m}$ for the streets and nodes $b_{1}, \ldots, b_{n}$ for the intersections. There are directed arcs from $A$ to each and from each $b_{i}$ to $B$. From each $a_{i}$ there are two further arcs, directed towards $b_{j}$ and $b_{j^{\prime}}$ that correspond to endpoints of street $i .]$

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• Paper 4, Section II, 20H

  #Optimization

  2013

  Given real numbers $a$ and $b$, consider the problem $\mathrm{P}$ of minimizing

  $$f(x)=a x_{11}+2 x_{12}+3 x_{13}+b x_{21}+4 x_{22}+x_{23}$$

  subject to $x_{i j} \geqslant 0$ and

  $$\begin{array}{r} x_{11}+x_{12}+x_{13}=5 \\ x_{21}+x_{22}+x_{23}=5 \\ x_{11}+x_{21}=3 \\ x_{12}+x_{22}=3 \\ x_{13}+x_{23}=4 \end{array}$$

  List all the basic feasible solutions, writing them as $2 \times 3$ matrices $\left(x_{i j}\right)$.

  Let $f(x)=\sum_{i j} c_{i j} x_{i j}$. Suppose there exist $\lambda_{i}, \mu_{j}$ such that

  $$\lambda_{i}+\mu_{j} \leqslant c_{i j} \text { for all } i \in\{1,2\}, j \in\{1,2,3\}$$

  Prove that if $x$ and $x^{\prime}$ are both feasible for $\mathrm{P}$ and $\lambda_{i}+\mu_{j}=c_{i j}$ whenever $x_{i j}>0$, then $f(x) \leqslant f\left(x^{\prime}\right) .$

  Let $x^{*}$ be the initial feasible solution that is obtained by formulating $\mathrm{P}$ as a transportation problem and using a greedy method that starts in the upper left of the matrix $\left(x_{i j}\right)$. Show that if $a+2 \leqslant b$ then $x^{*}$ minimizes $f$.

  For what values of $a$ and $b$ is one step of the transportation algorithm sufficient to pivot from $x^{*}$ to a solution that maximizes $f$ ?

  comment

• Paper 3, Section II, H

  #Optimization

  2013

  Use the two phase method to find all optimal solutions to the problem

  $$\begin{aligned} \operatorname{maximize} 2 x_{1}+3 x_{2}+x_{3} \\ \text { subject to } x_{1}+x_{2}+x_{3} & \leqslant 40 \\ 2 x_{1}+x_{2}-x_{3} & \geqslant 10 \\ -x_{2}+x_{3} & \geqslant 10 \\ x_{1}, x_{2}, x_{3} & \geqslant 0 \end{aligned}$$

  Suppose that the values $(40,10,10)$ are perturbed to $(40,10,10)+\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)$. Find an expression for the change in the optimal value, which is valid for all sufficiently small values of $\epsilon_{1}, \epsilon_{2}, \epsilon_{3}$.

  Suppose that $\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)=(\theta,-2 \theta, 0)$. For what values of $\theta$ is your expression valid?

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• Paper 4, Section I, B

  #Quantum Mechanics

  2013

  The components of the three-dimensional angular momentum operator $\hat{\mathbf{L}}$ are defined as follows:

  $$\hat{L}_{x}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right) \quad \hat{L}_{y}=-i \hbar\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right) \quad \hat{L}_{z}=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)$$

  Given that the wavefunction

  $$\psi=(f(x)+i y) z$$

  is an eigenfunction of $\hat{L}_{z}$, find all possible values of $f(x)$ and the corresponding eigenvalues of $\psi$. Letting $f(x)=x$, show that $\psi$ is an eigenfunction of $\hat{\mathbf{L}}^{2}$ and calculate the corresponding eigenvalue.

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• Paper 3, Section I, B

  #Quantum Mechanics

  2013

  If $\alpha, \beta$ and $\gamma$ are linear operators, establish the identity

  $$[\alpha \beta, \gamma]=\alpha[\beta, \gamma]+[\alpha, \gamma] \beta$$

  In what follows, the operators $x$ and $p$ are Hermitian and represent position and momentum of a quantum mechanical particle in one-dimension. Show that

  $$\left[x^{n}, p\right]=i \hbar n x^{n-1}$$

  and

  $$\left[x, p^{m}\right]=i \hbar m p^{m-1}$$

  where $m, n \in \mathbb{Z}^{+}$. Assuming $\left[x^{n}, p^{m}\right] \neq 0$, show that the operators $x^{n}$ and $p^{m}$ are Hermitian but their product is not. Determine whether $x^{n} p^{m}+p^{m} x^{n}$ is Hermitian.

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• Paper 1, Section II, B

  #Quantum Mechanics

  2013

  A particle with momentum $\hat{p}$ moves in a one-dimensional real potential with Hamiltonian given by

  $$\hat{H}=\frac{1}{2 m}(\hat{p}+i s A)(\hat{p}-i s A), \quad-\infty<x<\infty$$

  where $A$ is a real function and $s \in \mathbb{R}^{+}$. Obtain the potential energy of the system. Find $\chi(x)$ such that $(\hat{p}-i s A) \chi(x)=0$. Now, putting $A=x^{n}$, for $n \in \mathbb{Z}^{+}$, show that $\chi(x)$ can be normalised only if $n$ is odd. Letting $n=1$, use the inequality

  $$\int_{-\infty}^{\infty} \psi^{*}(x) \hat{H} \psi(x) d x \geqslant 0$$

  to show that

  $$\Delta x \Delta p \geqslant \frac{\hbar}{2}$$

  assuming that both $\langle\hat{p}\rangle$ and $\langle\hat{x}\rangle$ vanish.

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• Paper 3, Section II, B

  #Quantum Mechanics

  2013

  Obtain, with the aid of the time-dependent Schrödinger equation, the conservation equation

  $$\frac{\partial}{\partial t} \rho(\mathbf{x}, t)+\nabla \cdot \mathbf{j}(\mathbf{x}, t)=0$$

  where $\rho(\mathbf{x}, t)$ is the probability density and $\mathbf{j}(\mathbf{x}, t)$ is the probability current. What have you assumed about the potential energy of the system?

  Show that if the potential $U(\mathbf{x}, t)$ is complex the conservation equation becomes

  $$\frac{\partial}{\partial t} \rho(\mathbf{x}, t)+\nabla \cdot \mathbf{j}(\mathbf{x}, t)=\frac{2}{\hbar} \rho(\mathbf{x}, t) \operatorname{Im} U(\mathbf{x}, t)$$

  Take the potential to be time-independent. Show, with the aid of the divergence theorem, that

  $$\frac{d}{d t} \int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) d V=\frac{2}{\hbar} \int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) \operatorname{Im} U(\mathbf{x}) d V$$

  Assuming the wavefunction $\psi(\mathbf{x}, 0)$ is normalised to unity, show that if $\rho(\mathbf{x}, t)$ is expanded about $t=0$ so that $\rho(\mathbf{x}, t)=\rho_{0}(\mathbf{x})+t \rho_{1}(\mathbf{x})+\cdots$, then

  $$\int_{\mathbb{R}^{3}} \rho(\mathbf{x}, t) d V=1+\frac{2 t}{\hbar} \int_{\mathbb{R}^{3}} \rho_{0}(\mathbf{x}) \operatorname{Im} U(\mathbf{x}) d V+\cdots$$

  As time increases, how does the quantity on the left of this equation behave if $\operatorname{Im} U(\mathbf{x})<0$ ?

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• Paper 2, Section II, B

  #Quantum Mechanics

  2013

  (i) Consider a particle of mass $m$ confined to a one-dimensional potential well of depth $U>0$ and potential

  $$V(x)=\left\{\begin{array}{cl} -U, & |x|<l \\ 0, & |x|>l \end{array}\right.$$

  If the particle has energy $E$ where $-U \leqslant E<0$, show that for even states

  $$\alpha \tan \alpha l=\beta$$

  where $\alpha=\left[\frac{2 m}{\hbar^{2}}(U+E)\right]^{1 / 2}$ and $\beta=\left[-\frac{2 m}{\hbar^{2}} E\right]^{1 / 2}$.

  (ii) A particle of mass $m$ that is incident from the left scatters off a one-dimensional potential given by

  $$V(x)=k \delta(x)$$

  where $\delta(x)$ is the Dirac delta. If the particle has energy $E>0$ and $k>0$, obtain the reflection and transmission coefficients $R$ and $T$, respectively. Confirm that $R+T=1$.

  For the case $k<0$ and $E<0$ show that the energy of the only even parity bound state of the system is given by

  $$E=-\frac{m k^{2}}{2 \hbar^{2}}$$

  Use part (i) to verify this result by taking the limit $U \rightarrow \infty, l \rightarrow 0$ with $U l$ fixed.

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• Paper 1, Section I, H

  #Statistics

  2013

  Let $x_{1}, \ldots, x_{n}$ be independent and identically distributed observations from a distribution with probability density function

  $$f(x)= \begin{cases}\lambda e^{-\lambda(x-\mu)}, & x \geqslant \mu \\ 0, & x<\mu\end{cases}$$

  where $\lambda$ and $\mu$ are unknown positive parameters. Let $\beta=\mu+1 / \lambda$. Find the maximum likelihood estimators $\hat{\lambda}, \hat{\mu}$ and $\hat{\beta}$.

  Determine for each of $\hat{\lambda}, \hat{\mu}$ and $\hat{\beta}$ whether or not it has a positive bias.

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• Paper 2, Section I, H

  #Statistics

  2013

  State and prove the Rao-Blackwell theorem.

  Individuals in a population are independently of three types $\{0,1,2\}$, with unknown probabilities $p_{0}, p_{1}, p_{2}$ where $p_{0}+p_{1}+p_{2}=1$. In a random sample of $n$ people the $i$ th person is found to be of type $x_{i} \in\{0,1,2\}$.

  Show that an unbiased estimator of $\theta=p_{0} p_{1} p_{2}$ is

  $$\hat{\theta}= \begin{cases}1, & \text { if }\left(x_{1}, x_{2}, x_{3}\right)=(0,1,2) \\ 0, & \text { otherwise. }\end{cases}$$

  Suppose that $n_{i}$ of the individuals are of type $i$. Find an unbiased estimator of $\theta$, say $\theta^{*}$, such that $\operatorname{var}\left(\theta^{*}\right)<\theta(1-\theta)$.

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• Paper 4, Section II, H

  #Statistics

  2013

  Explain the notion of a sufficient statistic.

  Suppose $X$ is a random variable with distribution $F$ taking values in $\{1, \ldots, 6\}$, with $P(X=i)=p_{i}$. Let $x_{1}, \ldots, x_{n}$ be a sample from $F$. Suppose $n_{i}$ is the number of these $x_{j}$ that are equal to $i$. Use a factorization criterion to explain why $\left(n_{1}, \ldots, n_{6}\right)$ is sufficient for $\theta=\left(p_{1}, \ldots, p_{6}\right)$.

  Let $H_{0}$ be the hypothesis that $p_{i}=1 / 6$ for all $i$. Derive the statistic of the generalized likelihood ratio test of $H_{0}$ against the alternative that this is not a good fit.

  Assuming that $n_{i} \approx n / 6$ when $H_{0}$ is true and $n$ is large, show that this test can be approximated by a chi-squared test using a test statistic

  $$T=-n+\frac{6}{n} \sum_{i=1}^{6} n_{i}^{2}$$

  Suppose $n=100$ and $T=8.12$. Would you reject $H_{0} ?$ Explain your answer.

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• Paper 1, Section II, H

  #Statistics

  2013

  Consider the general linear model $Y=X \theta+\epsilon$ where $X$ is a known $n \times p$ matrix, $\theta$ is an unknown $p \times 1$ vector of parameters, and $\epsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with unknown variance $\sigma^{2}$. Assume the $p \times p$ matrix $X^{T} X$ is invertible. Let

  $$\begin{aligned} \hat{\theta} &=\left(X^{T} X\right)^{-1} X^{T} Y \\ \hat{\epsilon} &=Y-X \hat{\theta} \end{aligned}$$

  What are the distributions of $\hat{\theta}$ and $\hat{\epsilon}$ ? Show that $\hat{\theta}$ and $\hat{\epsilon}$ are uncorrelated.

  Four apple trees stand in a $2 \times 2$ rectangular grid. The annual yield of the tree at coordinate $(i, j)$ conforms to the model

  $$y_{i j}=\alpha_{i}+\beta x_{i j}+\epsilon_{i j}, \quad i, j \in\{1,2\},$$

  where $x_{i j}$ is the amount of fertilizer applied to tree $(i, j), \alpha_{1}, \alpha_{2}$ may differ because of varying soil across rows, and the $\epsilon_{i j}$ are $N\left(0, \sigma^{2}\right)$ random variables that are independent of one another and from year to year. The following two possible experiments are to be compared:

  $$\mathrm{I}:\left(x_{i j}\right)=\left(\begin{array}{cc} 0 & 1 \\ 2 & 3 \end{array}\right) \quad \text { and } \quad \mathrm{II}:\left(x_{i j}\right)=\left(\begin{array}{cc} 0 & 2 \\ 3 & 1 \end{array}\right) \text {. }$$

  Represent these as general linear models, with $\theta=\left(\alpha_{1}, \alpha_{2}, \beta\right)$. Compare the variances of estimates of $\beta$ under I and II.

  With II the following yields are observed:

  $$\left(y_{i j}\right)=\left(\begin{array}{ll} 100 & 300 \\ 600 & 400 \end{array}\right)$$

  Forecast the total yield that will be obtained next year if no fertilizer is used. What is the $95 \%$ predictive interval for this yield?

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• Paper 3, Section II, H

  #Statistics

  2013

  Suppose $x_{1}$ is a single observation from a distribution with density $f$ over $[0,1]$. It is desired to test $H_{0}: f(x)=1$ against $H_{1}: f(x)=2 x$.

  Let $\delta:[0,1] \rightarrow\{0,1\}$ define a test by $\delta\left(x_{1}\right)=i \Longleftrightarrow$ 'accept $H_{i}$ '. Let $\alpha_{i}(\delta)=P\left(\delta\left(x_{1}\right)=1-i \mid H_{i}\right)$. State the Neyman-Pearson lemma using this notation.

  Let $\delta$ be the best test of size $0.10$. Find $\delta$ and $\alpha_{1}(\delta)$.

  Consider now $\delta:[0,1] \rightarrow\{0,1, \star\}$ where $\delta\left(x_{1}\right)=\star$ means 'declare the test to be inconclusive'. Let $\gamma_{i}(\delta)=P\left(\delta(x)=\star \mid H_{i}\right)$. Given prior probabilities $\pi_{0}$ for $H_{0}$ and $\pi_{1}=1-\pi_{0}$ for $H_{1}$, and some $w_{0}, w_{1}$, let

  $$\operatorname{cost}(\delta)=\pi_{0}\left(w_{0} \alpha_{0}(\delta)+\gamma_{0}(\delta)\right)+\pi_{1}\left(w_{1} \alpha_{1}(\delta)+\gamma_{1}(\delta)\right)$$

  Let $\delta^{*}\left(x_{1}\right)=i \Longleftrightarrow x_{1} \in A_{i}$, where $A_{0}=[0,0.5), A_{\star}=[0.5,0.6), A_{1}=[0.6,1]$. Prove that for each value of $\pi_{0} \in(0,1)$ there exist $w_{0}, w_{1}$ (depending on $\left.\pi_{0}\right)$ such that $\operatorname{cost}\left(\delta^{*}\right)=\min _{\delta} \operatorname{cost}(\delta) .\left[\right.$ Hint $\left.: w_{0}=1+2(0.6)\left(\pi_{1} / \pi_{0}\right) .\right]$

  Hence prove that if $\delta$ is any test for which

  $$\alpha_{i}(\delta) \leqslant \alpha_{i}\left(\delta^{*}\right), \quad i=0,1$$

  then $\gamma_{0}(\delta) \geqslant \gamma_{0}\left(\delta^{*}\right)$ and $\gamma_{1}(\delta) \geqslant \gamma_{1}\left(\delta^{*}\right)$.

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• Paper 1, Section I, A

  #Variational Principles

  2013

  (a) Define what it means for a function $g: \mathbb{R} \rightarrow \mathbb{R}$ to be convex. Assuming $g^{\prime \prime}$ exists, state an equivalent condition. Let $f(x)=x \log x$, defined on $x>0$. Show that $f(x)$ is convex.

  (b) Find the Legendre transform $f^{*}(p)$ of $f(x)=x \log x$. State the domain of $f^{*}(p)$. Without further calculation, explain why $\left(f^{*}\right)^{*}=f$ in this case.

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• Paper 3, Section II, H

  #Algebraic Geometry

  2013

  Let $C \subset \mathbb{P}^{2}$ be the plane curve given by the polynomial

  $$X_{0}^{n}-X_{1}^{n}-X_{2}^{n}$$

  over the field of complex numbers, where $n \geqslant 3$.

  (i) Show that $C$ is nonsingular.

  (ii) Compute the divisors of the rational functions

  $$x=\frac{X_{1}}{X_{0}}, \quad y=\frac{X_{2}}{X_{0}}$$

  on $C$.

  (iii) Consider the morphism $\phi=\left(X_{0}: X_{1}\right): C \rightarrow \mathbb{P}^{1}$. Compute its ramification points and degree.

  (iv) Show that a basis for the space of regular differentials on $C$ is

  $$\left\{x^{i} y^{j} \omega_{0} \mid i, j \geqslant 0, i+j \leqslant n-3\right\}$$

  where $\omega_{0}=d x / y^{n-1} .$

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• Paper 4, Section II, H

  #Algebraic Geometry

  2013

  Let $C$ be a nonsingular projective curve, and $D$ a divisor on $C$ of degree $d$.

  (i) State the Riemann-Roch theorem for $D$, giving a brief explanation of each term. Deduce that if $d>2 g-2$ then $\ell(D)=1-g+d$.

  (ii) Show that, for every $P \in C$,

  $$\ell(D-P) \geqslant \ell(D)-1$$

  Deduce that $\ell(D) \leqslant 1+d$. Show also that if $\ell(D)>1$, then $\ell(D-P)=\ell(D)-1$ for all but finitely many $P \in C$.

  (iii) Deduce that for every $d \geqslant g-1$ there exists a divisor $D$ of degree $d$ with $\ell(D)=1-g+d$.

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• Paper 2, Section II, H

  #Algebraic Geometry

  2013

  Let $V \subset \mathbb{P}^{3}$ be an irreducible quadric surface.

  (i) Show that if $V$ is singular, then every nonsingular point lies in exactly one line in $V$, and that all the lines meet in the singular point, which is unique.

  (ii) Show that if $V$ is nonsingular then each point of $V$ lies on exactly two lines of $V$.

  Let $V$ be nonsingular, $P_{0}$ a point of $V$, and $\Pi \subset \mathbb{P}^{3}$ a plane not containing $P_{0}$. Show that the projection from $P_{0}$ to $\Pi$ is a birational map $f: V \rightarrow \rightarrow \Pi$. At what points does $f$ fail to be regular? At what points does $f^{-1}$ fail to be regular? Justify your answers.

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• Paper 1, Section II, H

  #Algebraic Geometry

  2013

  Let $V \subset \mathbb{A}^{n}$ be an affine variety over an algebraically closed field $k$. What does it mean to say that $V$ is irreducible? Show that any non-empty affine variety $V \subset \mathbb{A}^{n}$ is the union of a finite number of irreducible affine varieties $V_{j} \subset \mathbb{A}^{n}$.

  Define the ideal $I(V)$ of $V$. Show that $I(V)$ is a prime ideal if and only if $V$ is irreducible.

  Assume that the base field $k$ has characteristic zero. Determine the irreducible components of

  $$V\left(X_{1} X_{2}, X_{1} X_{3}+X_{2}^{2}-1, X_{1}^{2}\left(X_{1}-X_{3}\right)\right) \subset \mathbb{A}^{3}$$

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• Paper 3, Section II, G

  #Algebraic Topology

  2013

  (i) State, but do not prove, the Mayer-Vietoris theorem for the homology groups of polyhedra.

  (ii) Calculate the homology groups of the $n$-sphere, for every $n \geqslant 0$.

  (iii) Suppose that $a \geqslant 1$ and $b \geqslant 0$. Calculate the homology groups of the subspace $X$ of $\mathbb{R}^{a+b}$ defined by $\sum_{i=1}^{a} x_{i}^{2}-\sum_{j=a+1}^{a+b} x_{j}^{2}=1$.

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• Paper 4, Section II, G

  #Algebraic Topology

  2013

  (i) State, but do not prove, the Lefschetz fixed point theorem.

  (ii) Show that if $n$ is even, then for every map $f: S^{n} \rightarrow S^{n}$ there is a point $x \in S^{n}$ such that $f(x)=\pm x$. Is this true if $n$ is odd? [Standard results on the homology groups for the $n$-sphere may be assumed without proof, provided they are stated clearly.]

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• Paper 2, Section II, G

  #Algebraic Topology

  2013

  (i) State the Seifert-van Kampen theorem.

  (ii) Assuming any standard results about the fundamental group of a circle that you wish, calculate the fundamental group of the $n$-sphere, for every $n \geqslant 2$.

  (iii) Suppose that $n \geqslant 3$ and that $X$ is a path-connected topological $n$-manifold. Show that $\pi_{1}\left(X, x_{0}\right)$ is isomorphic to $\pi_{1}\left(X-\{P\}, x_{0}\right)$ for any $P \in X-\left\{x_{0}\right\}$.

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• Paper 1, Section II, 21G

  #Algebraic Topology

  2013

  (i) Define the notion of the fundamental group $\pi_{1}\left(X, x_{0}\right)$ of a path-connected space $X$ with base point $x_{0}$.

  (ii) Prove that if a group $G$ acts freely and properly discontinuously on a simply connected space $Z$, then $\pi_{1}\left(G \backslash Z, x_{0}\right)$ is isomorphic to $G$. [You may assume the homotopy lifting property, provided that you state it clearly.]

  (iii) Suppose that $p, q$ are distinct points on the 2 -sphere $S^{2}$ and that $X=S^{2} /(p \sim q)$. Exhibit a simply connected space $Z$ with an action of a group $G$ as in (ii) such that $X=G \backslash Z$, and calculate $\pi_{1}\left(X, x_{0}\right)$.

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• Paper 4, Section II, D

  #Applications of Quantum Mechanics

  2013

  Define the Floquet matrix for a particle moving in a periodic potential in one dimension and explain how it determines the allowed energy bands of the system.

  A potential barrier in one dimension has the form

  $$V(x)= \begin{cases}V_{0}(x), & |x|<a / 4 \\ 0, & |x|>a / 4\end{cases}$$

  where $V_{0}(x)$ is a smooth, positive function of $x$. The reflection and transmission amplitudes for a particle of wavenumber $k>0$, incident from the left, are $r(k)$ and $t(k)$ respectively. For a particle of wavenumber $-k$, incident from the right, the corresponding amplitudes are $r^{\prime}(k)$ and $t^{\prime}(k)=t(k)$. In the following, for brevity, we will suppress the $k$-dependence of these quantities.

  Consider the periodic potential $\tilde{V}$, defined by $\tilde{V}(x)=V(x)$ for $|x|<a / 2$ and by $\tilde{V}(x+a)=\widetilde{V}(x)$ elsewhere. Write down two linearly independent solutions of the corresponding Schrödinger equation in the region $-3 a / 4<x<-a / 4$. Using the scattering data given above, extend these solutions to the region $a / 4<x<3 a / 4$. Hence find the Floquet matrix of the system in terms of the amplitudes $r, r^{\prime}$ and $t$ defined above.

  Show that the edges of the allowed energy bands for this potential lie at $E=\hbar^{2} k^{2} / 2 m$, where

  $$k a=i \log \left(t \pm \sqrt{r r^{\prime}}\right)$$

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• Paper 2, Section II, D

  #Applications of Quantum Mechanics

  2013

  (i) A particle of momentum $\hbar k$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters off a sphericallysymmetric target in three dimensions. Define the corresponding scattering amplitude $f$ as a function of the scattering angle $\theta$. Expand the scattering amplitude in partial waves of definite angular momentum $l$, and determine the coefficients of this expansion in terms of the phase shifts $\delta_{l}(k)$ appearing in the following asymptotic form of the wavefunction, valid at large distance from the target,

  $$\psi(\mathbf{r}) \sim \sum_{l=0}^{\infty} \frac{2 l+1}{2 i k}\left[e^{2 i \delta_{l}} \frac{e^{i k r}}{r}-(-1)^{l} \frac{e^{-i k r}}{r}\right] P_{l}(\cos \theta) .$$

  Here, $r=|\mathbf{r}|$ is the distance from the target and $P_{l}$ are the Legendre polynomials.

  [You may use without derivation the following approximate relation between plane and spherical waves (valid asymptotically for large $r$ ):

  $$\exp (i k z) \sim \sum_{l=0}^{\infty}(2 l+1) i^{l} \frac{\sin \left(k r-\frac{1}{2} l \pi\right)}{k r} P_{l}(\cos \theta) .$$

  (ii) Suppose that the potential energy takes the form $V(r)=\lambda U(r)$ where $\lambda \ll 1$ is a dimensionless coupling. By expanding the wavefunction in a power series in $\lambda$, derive the Born Approximation to the scattering amplitude in the form

  $$f(\theta)=-\frac{2 m \lambda}{\hbar^{2}} \int_{0}^{\infty} U(r) \frac{\sin q r}{q} r d r$$

  up to corrections of order $\lambda^{2}$, where $q=2 k \sin (\theta / 2)$. [You may quote any results you need for the Green's function for the differential operator $\nabla^{2}+k^{2}$ provided they are stated clearly.]

  (iii) Derive the corresponding order $\lambda$ contribution to the phase shift $\delta_{l}(k)$ of angular momentum $l$.

  [You may use the orthogonality relations

  $$\int_{-1}^{+1} P_{l}(w) P_{m}(w) d w=\frac{2}{(2 l+1)} \delta_{l m}$$

  and the integral formula

  $$\int_{0}^{1} P_{l}\left(1-2 x^{2}\right) \sin (a x) d x=\frac{a}{2}\left[j_{l}\left(\frac{a}{2}\right)\right]^{2}$$

  where $j_{l}(z)$ is a spherical Bessel function.]

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• Paper 3, Section II, D

  #Applications of Quantum Mechanics

  2013

  Write down the classical Hamiltonian for a particle of mass $m$, electric charge $-e$ and momentum p moving in the background of an electromagnetic field with vector and scalar potentials $\mathbf{A}(\mathbf{x}, t)$ and $\phi(\mathbf{x}, t)$.

  Consider the case of a constant uniform magnetic field, $\mathbf{B}=(0,0, B)$ and $\mathbf{E}=0$. Working in the gauge with $\mathbf{A}=(-B y, 0,0)$ and $\phi=0$, show that Hamilton's equations,

  $$\dot{\mathbf{x}}=\frac{\partial H}{\partial \mathbf{p}}, \quad \dot{\mathbf{p}}=-\frac{\partial H}{\partial \mathbf{x}},$$

  admit solutions corresponding to circular motion in the $x-y$ plane with angular frequency $\omega_{B}=e B / m$.

  Show that, in the same gauge, the coordinates $\left(x_{0}, y_{0}, 0\right)$ of the centre of the circle are related to the instantaneous position $\mathbf{x}=(x, y, z)$ and momentum $\mathbf{p}=\left(p_{x}, p_{y}, p_{z}\right)$ of the particle by

  $$x_{0}=x-\frac{p_{y}}{e B}, \quad y_{0}=\frac{p_{x}}{e B} .$$

  Write down the quantum Hamiltonian $\hat{H}$ for the system. In the case of a uniform constant magnetic field discussed above, find the allowed energy levels. Working in the gauge specified above, write down quantum operators corresponding to the classical quantities $x_{0}$ and $y_{0}$ defined in (1) above and show that they are conserved.

  [In this question you may use without derivation any facts relating to the energy spectrum of the quantum harmonic oscillator provided they are stated clearly.]

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• Paper 1, Section II, D

  #Applications of Quantum Mechanics

  2013

  Consider a quantum system with Hamiltonian $\widehat{H}$ and energy levels

  $$E_{0}<E_{1}<E_{2}<\ldots$$

  For any state $|\psi\rangle$ define the Rayleigh-Ritz quotient $R[\psi]$ and show the following:

  (i) the ground state energy $E_{0}$ is the minimum value of $R[\psi]$;

  (ii) all energy eigenstates are stationary points of $R[\psi]$ with respect to variations of $|\psi\rangle$.

  Under what conditions can the value of $R\left[\psi_{\alpha}\right]$ for a trial wavefunction $\psi_{\alpha}$ (depending on some parameter $\alpha$ ) be used as an estimate of the energy $E_{1}$ of the first excited state? Explain your answer.

  For a suitably chosen trial wavefunction which is the product of a polynomial and a Gaussian, use the Rayleigh-Ritz quotient to estimate $E_{1}$ for a particle of mass $m$ moving in a potential $V(x)=g|x|$, where $g$ is a constant.

  [You may use the integral formulae,

  $$\begin{aligned} \int_{0}^{\infty} x^{2 n} \exp \left(-p x^{2}\right) d x &=\frac{(2 n-1) ! !}{2(2 p)^{n}} \sqrt{\frac{\pi}{p}} \\ \int_{0}^{\infty} x^{2 n+1} \exp \left(-p x^{2}\right) d x &=\frac{n !}{2 p^{n+1}} \end{aligned}$$

  where $n$ is a non-negative integer and $p$ is a constant. ]

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• Paper 4, Section II, J

  #Applied Probability

  2013

  (i) Define an $M / M / 1$ queue. Justifying briefly your answer, specify when this queue has a stationary distribution, and identify that distribution. State and prove Burke's theorem for this queue.

  (ii) Let $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ denote a Jackson network of $N$ queues, where the entrance and service rates for queue $i$ are respectively $\lambda_{i}$ and $\mu_{i}$, and each customer leaving queue $i$ moves to queue $j$ with probability $p_{i j}$ after service. We assume $\sum_{j} p_{i j}<1$ for each $i=1, \ldots, N$; with probability $1-\sum_{j} p_{i j}$ a customer leaving queue $i$ departs from the system. State Jackson's theorem for this network. [You are not required to prove it.] Are the processes $\left(L_{1}(t), \ldots, L_{N}(t), t \geqslant 0\right)$ independent at equilibrium? Justify your answer.

  (iii) Let $D_{i}(t)$ be the process of final departures from queue $i$. Show that, at equilibrium, $\left(L_{1}(t), \ldots, L_{N}(t)\right)$ is independent of $\left(D_{i}(s), 1 \leqslant i \leqslant N, 0 \leqslant s \leqslant t\right)$. Show that, for each fixed $i=1, \ldots, N,\left(D_{i}(t), t \geqslant 0\right)$ is a Poisson process, and specify its rate.

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• Paper 3, Section II, J

  #Applied Probability

  2013

  Define the Moran model. Describe briefly the infinite sites model of mutations.

  We henceforth consider a population with $N$ individuals evolving according to the rules of the Moran model. In addition we assume:

  • the allelic type of any individual at any time lies in a given countable state space $S$;

  • individuals are subject to mutations at constant rate $u=\theta / N$, independently of the population dynamics;

  • each time a mutation occurs, if the allelic type of the individual was $x \in S$, it changes to $y \in S$ with probability $P(x, y)$, where $P(x, y)$ is a given Markovian transition matrix on $S$ that is symmetric:

  $$P(x, y)=P(y, x) \quad(x, y \in S)$$

  (i) Show that, if two individuals are sampled at random from the population at some time $t$, then the time to their most recent common ancestor has an exponential distribution, with a parameter that you should specify.

  (ii) Let $\Delta+1$ be the total number of mutations that accumulate on the two branches separating these individuals from their most recent common ancestor. Show that $\Delta+1$ is a geometric random variable, and specify its probability parameter $p$.

  (iii) The first individual is observed to be of type $x \in S$. Explain why the probability that the second individual is also of type $x$ is

  $$\mathbb{P}\left(X_{\Delta}=x \mid X_{0}=x\right),$$

  where $\left(X_{n}, n \geqslant 0\right)$ is a Markov chain on $S$ with transition matrix $P$ and is independent of $\Delta$.

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• Paper 2, Section II, J

  #Applied Probability

  2013

  (i) Define a Poisson process as a Markov chain on the non-negative integers and state three other characterisations.

  (ii) Let $\lambda(s)(s \geqslant 0)$ be a continuous positive function. Let $\left(X_{t}, t \geqslant 0\right)$ be a right-continuous process with independent increments, such that

  $$\begin{aligned} \mathbb{P}\left(X_{t+h}=X_{t}+1\right) &=\lambda(t) h+o(h) \\ \mathbb{P}\left(X_{t+h}=X_{t}\right) &=1-\lambda(t) h+o(h) \end{aligned}$$

  where the $o(h)$ terms are uniform in $t \in[0, \infty)$. Show that $X_{t}$ is a Poisson random variable with parameter $\Lambda(t)=\int_{0}^{t} \lambda(s) d s$.

  (iii) Let $X=\left(X_{n}: n=1,2, \ldots\right)$ be a sequence of independent and identically distributed positive random variables with continuous density function $f$. We define the sequence of successive records, $\left(K_{n}, n=0,1, \ldots\right)$, by $K_{0}:=0$ and, for $n \geqslant 0$,

  $$K_{n+1}:=\inf \left\{m>K_{n}: X_{m}>X_{K_{n}}\right\}$$

  The record process,$\left(R_{t}, t \geqslant 0\right)$, is then defined by

  $$R_{t}:=\#\left\{n \geqslant 1: X_{K_{n}} \leqslant t\right\}$$

  Explain why the increments of $R$ are independent. Show that $R_{t}$ is a Poisson random variable with parameter $-\log \{1-F(t)\}$ where $F(t)=\int_{0}^{t} f(s) d s$.

  [You may assume the following without proof: For fixed $t>0$, let $Y$ (respectively, $Z$ ) be the subsequence of $X$ obtained by retaining only those elements that are greater than (respectively, smaller than) $t$. Then $Y$ (respectively, $Z$ ) is a sequence of independent variables each having the distribution of $X_{1}$ conditioned on $X_{1}>t$ (respectively, $X_{1}<t$ ); and $Y$ and $Z$ are independent.]

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• Paper 1, Section II, J

  #Applied Probability

  2013

  Let $\left(X_{t}, t \geqslant 0\right)$ be a Markov chain on $\{0,1, \ldots\}$ with $Q$-matrix given by

  $$\begin{aligned} q_{n, n+1} &=\lambda_{n} \\ q_{n, 0} &=\lambda_{n} \varepsilon_{n} \quad(n>0) \\ q_{n, m} &=0 \quad \text { if } m \notin\{0, n, n+1\} \end{aligned}$$

  where $\varepsilon_{n}, \lambda_{n}>0$.

  (i) Show that $X$ is transient if and only if $\sum_{n} \varepsilon_{n}<\infty$. [You may assume without proof that $x(1-\delta) \leqslant \log (1+x) \leqslant x$ for all $\delta>0$ and all sufficiently small positive $x$.]

  (ii) Assume that $\sum_{n} \varepsilon_{n}<\infty$. Find a necessary and sufficient condition for $X$ to be almost surely explosive. [You may assume without proof standard results about pure birth processes, provided that they are stated clearly.]

  (iii) Find a stationary measure for $X$. For the case $\lambda_{n}=\lambda$ and $\varepsilon_{n}=\alpha /(n+1)(\lambda, \alpha>0)$, show that $X$ is positive recurrent if and only if $\alpha>1$.

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• Paper 4, Section II, B

  #Asymptotic Methods

  2013

  Show that the equation

  $$\frac{d^{2} y}{d x^{2}}+\frac{2}{x} \frac{d y}{d x}+\left(\frac{1}{x^{2}}-1\right) y=0$$

  has an irregular singular point at infinity. Using the Liouville-Green method, show that one solution has the asymptotic expansion

  $$y(x) \sim \frac{1}{x} e^{x}\left(1+\frac{1}{2 x}+\ldots\right)$$

  as $x \rightarrow \infty$

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• Paper 3, Section II, B

  #Asymptotic Methods

  2013

  Let

  $$I(x)=\int_{0}^{\pi} f(t) e^{i x \psi(t)} d t$$

  where $f(t)$ and $\psi(t)$ are smooth, and $\psi^{\prime}(t) \neq 0$ for $t>0 ;$ also $f(0) \neq 0$, $\psi(0)=a$, $\psi^{\prime}(0)=\psi^{\prime \prime}(0)=0$ and $\psi^{\prime \prime \prime}(0)=6 b>0$. Show that, as $x \rightarrow+\infty$,

  $$I(x) \sim f(0) e^{i(x a+\pi / 6)}\left(\frac{1}{27 b x}\right)^{1 / 3} \Gamma(1 / 3) .$$

  Consider the Bessel function

  $$J_{n}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (n t-x \sin t) d t$$

  Show that, as $n \rightarrow+\infty$,

  $$J_{n}(n) \sim \frac{\Gamma(1 / 3)}{\pi} \frac{1}{(48)^{1 / 6}} \frac{1}{n^{1 / 3}}$$

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• Paper 1, Section II, B

  #Asymptotic Methods

  2013

  Suppose $\alpha>0$. Define what it means to say that

  $$F(x) \sim \frac{1}{\alpha x} \sum_{n=0}^{\infty} n !\left(\frac{-1}{\alpha x}\right)^{n}$$

  is an asymptotic expansion of $F(x)$ as $x \rightarrow \infty$. Show that $F(x)$ has no other asymptotic expansion in inverse powers of $x$ as $x \rightarrow \infty$.

  To estimate the value of $F(x)$ for large $x$, one may use an optimal truncation of the asymptotic expansion. Explain what is meant by this, and show that the error is an exponentially small quantity in $x$.

  Derive an integral respresentation for a function $F(x)$ with the above asymptotic expansion.

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• Paper 4, Section I, B

  #Classical Dynamics

  2013

  The Lagrangian for a heavy symmetric top of mass $M$, pinned at point $O$ which is a distance $l$ from the centre of mass, is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (i) Starting with the fixed space frame $\left(\tilde{\mathbf{e}}_{\mathbf{1}}, \tilde{\mathbf{e}}_{2}, \tilde{\mathbf{e}}_{3}\right)$ and choosing $O$ at its origin, sketch the top with embedded body frame axis $\mathbf{e}_{3}$ being the symmetry axis. Clearly identify the Euler angles $(\theta, \phi, \psi)$.

  (ii) Obtain the momenta $p_{\theta}, p_{\phi}$ and $p_{\psi}$ and the Hamiltonian $H\left(\theta, \phi, \psi, p_{\theta}, p_{\phi}, p_{\psi}\right)$. Derive Hamilton's equations. Identify the three conserved quantities.

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• Paper 3, Section I, B

  #Classical Dynamics

  2013

  Two equal masses $m$ are connected to each other and to fixed points by three springs of force constant $5 k, k$ and $5 k$ as shown in the figure.

  

  (i) Write down the Lagrangian and derive the equations describing the motion of the system in the direction parallel to the springs.

  (ii) Find the normal modes and their frequencies. Comment on your results.

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• Paper 2, Section I, B

  #Classical Dynamics

  2013

  (i) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}, I_{3}$. Derive Euler's equations of torque-free motion,

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}, \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1}, \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}, \end{aligned}$$

  with components of the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ given in the body frame.

  (ii) Use Euler's equations to show that the energy $E$ and the square of the total angular momentum $\mathbf{L}^{2}$ of the body are conserved.

  (iii) Consider a torque-free motion of a symmetric top with $I_{1}=I_{2}=\frac{1}{2} I_{3}$. Show that in the body frame the vector of angular velocity $\boldsymbol{\omega}$ precesses about the body-fixed $\mathbf{e}_{3}$ axis with constant angular frequency equal to $\omega_{3}$.

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• Paper 1, Section I, B

  #Classical Dynamics

  2013

  Consider an $n$-dimensional dynamical system with generalized coordinates and momenta $\left(q_{i}, p_{i}\right), i=1,2, \ldots, n$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right)$ and $g\left(q_{i}, p_{i}, t\right)$.

  (b) Assuming Hamilton's equations of motion, prove that if a function $G\left(q_{i}, p_{i}\right)$ Poisson commutes with the Hamiltonian, that is $\{G, H\}=0$, then $G$ is a constant of the motion.

  (c) Assume that $q_{j}$ is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum $p_{j}$ is conserved.

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• Paper 4, Section II, B

  #Classical Dynamics

  2013

  The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

  $$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  (i) Write down the Hamiltonian of the particle.

  (ii) Write down Hamilton's equations of motion for the particle.

  (iii) Show that Hamilton's equations are invariant under the gauge transformation

  $$\phi \rightarrow \phi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda,$$

  for an arbitrary function $\Lambda(\mathbf{r}, t)$.

  (iv) The particle moves in the presence of a field such that $\phi=0$ and $\mathbf{A}=\left(-\frac{1}{2} y B, \frac{1}{2} x B, 0\right)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant.

  (a) Find a gauge transformation such that only one component of $\mathbf{A}(x, y, z)$ remains non-zero.

  (b) Determine the motion of the particle.

  (v) Now assume that $B$ varies very slowly with time on a time-scale much longer than $(q B / m)^{-1}$. Find the quantity which remains approximately constant throughout the motion.

  [You may use the expression for the action variable $I=\frac{1}{2 \pi} \oint p_{i} d q_{i}$.]

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• Paper 2, Section II, B

  #Classical Dynamics

  2013

  (i) The action for a system with a generalized coordinate $q$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  (a) State the Principle of Least Action and derive the Euler-Lagrange equation.

  (b) Consider an arbitrary function $f(q, t)$. Show that $L^{\prime}=L+d f / d t$ leads to the same equation of motion.

  (ii) A wire frame $A B C$ in a shape of an equilateral triangle with side $a$ rotates in a horizontal plane with constant angular frequency $\omega$ about a vertical axis through $A$. A bead of mass $m$ is threaded on $B C$ and moves without friction. The bead is connected to $B$ and $C$ by two identical light springs of force constant $k$ and equilibrium length $a / 2$.

  (a) Introducing the displacement $\eta$ of the particle from the mid point of $B C$, determine the Lagrangian $L(\eta, \dot{\eta})$.

  (b) Derive the equation of motion. Identify the integral of the motion.

  (c) Describe the motion of the bead. Find the condition for there to be a stable equilibrium and find the frequency of small oscillations about it when it exists.

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• Paper 4, Section I, $4 \mathrm{H}$

  #Coding and Cryptography

  2013

  Describe how a stream cipher works. What is a one-time pad?

  A one-time pad is used to send the message $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} y_{7}$ which is encoded as 0101011. In error, it is reused to send the message $y_{0} x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ which is encoded as 0100010 . Show that there are two possibilities for the substring $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$, and find them.

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• Paper 3, Section I, H

  #Coding and Cryptography

  2013

  Describe briefly the Rabin cipher with modulus $N$, explaining how it can be deciphered by the intended recipient and why it is difficult for an eavesdropper to decipher it.

  The Cabinet decides to communicate using Rabin ciphers to maintain confidentiality. The Cabinet Secretary encrypts a message, represented as a positive integer $m$, using the Rabin cipher with modulus $N$ (with $0<m<N$ ) and publishes both the encrypted message and the modulus. The Defence Secretary deciphers this message to read it but then foolishly encrypts it again using a Rabin cipher with a different modulus $N^{\prime}$ (with $\left.m<N^{\prime}\right)$ and publishes the newly encrypted message and $N^{\prime}$. Mr Rime (the Leader of the Opposition) knows this has happened. Explain how Rime can work out what the original message was using the two different encrypted versions.

  Can Rime decipher other messages sent out by the Cabinet using the original modulus $N$ ?

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• Paper 2, Section I, H

  #Coding and Cryptography

  2013

  Let $A(n, d)$ denote the maximum size of a binary code of length $n$ with minimum distance $d$. For fixed $\delta$ with $0<\delta<1 / 2$, let $\alpha(\delta)=\limsup _{n} \frac{1}{n} \log _{2} A(n, n \delta)$. Show that

  $$1-H(\delta) \leqslant \alpha(\delta) \leqslant 1-H(\delta / 2)$$

  where $H(p)=-p \log _{2} p-(1-p) \log _{2}(1-p)$.

  [You may assume the GSV and Hamming bounds and any form of Stirling's theorem provided you state them clearly.]

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• Paper 1, Section I, H

  #Coding and Cryptography

  2013

  A binary Huffman code is used for encoding symbols $1, \ldots, m$ occurring with respective probabilities $p_{1} \geqslant \cdots \geqslant p_{m}>0$ where $\sum_{1 \leqslant j \leqslant m} p_{j}=1$. Let $s_{1}$ be the length of a shortest codeword and $s_{m}$ the length of a longest codeword. Determine the maximal and minimal values of each of $s_{1}$ and $s_{m}$, and find binary trees for which they are attained.

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• Paper 2, Section II, H

  #Coding and Cryptography

  2013

  Define a BCH code of length $n$, where $n$ is odd, over the field of 2 elements with design distance $\delta$. Show that the minimum weight of such a code is at least $\delta$. [Results about the van der Monde determinant may be quoted without proof, provided they are stated clearly.]

  Consider a BCH code of length 31 over the field of 2 elements with design distance 8 . Show that the minimum distance is at least 11. [Hint: Let $\alpha$ be a primitive element in the field of $2^{5}$ elements, and consider the minimal polynomial for certain powers of $\left.\alpha .\right]$

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• Paper 1, Section II, H

  #Coding and Cryptography

  2013

  Define the bar product $C_{1} \mid C_{2}$ of binary linear codes $C_{1}$ and $C_{2}$, where $C_{2}$ is a subcode of $C_{1}$. Relate the rank and minimum distance of $C_{1} \mid C_{2}$ to those of $C_{1}$ and $C_{2}$ and justify your answer. Show that if $C^{\perp}$ denotes the dual code of $C$, then

  $$\left(C_{1} \mid C_{2}\right)^{\perp}=C_{2}^{\perp} \mid C_{1}^{\perp}$$

  Using the bar product construction, or otherwise, define the Reed-Muller code $\mathrm{RM}(d, r)$ for $0 \leqslant r \leqslant d$. Show that if $0 \leqslant r \leqslant d-1$, then the dual of $\mathrm{RM}(d, r)$ is again a Reed-Muller code.

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• Paper 4, Section I, D

  #Cosmology

  2013

  List the relativistic species of bosons and fermions from the standard model of particle physics that are present in the early universe when the temperature falls to $1 \mathrm{MeV} / \mathrm{k}_{B}$.

  Which of the particles above will be interacting when the temperature is above $1 \mathrm{MeV} / k_{B}$ and between $1 \mathrm{MeV} / k_{B} \geq T \geq 0.51 \mathrm{MeV} / k_{B}$, respectively?

  Explain what happens to the populations of particles present when the temperature falls to $0.51 \mathrm{MeV} / k_{B}$.

  The entropy density of fermion and boson species with temperature $T$ is $s \propto g_{s} T^{3}$, where $g_{s}$ is the number of relativistic spin degrees of freedom, that is,

  $$g_{s}=\sum_{\text {bosons }} g_{i}+\frac{7}{8} \sum_{\text {fermions }} g_{i}$$

  Show that when the temperature of the universe falls below $0.51 \mathrm{MeV} / k_{B}$ the ratio of the neutrino and photon temperatures will be given by

  $$\frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1 / 3}$$

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• Paper 3, Section I, D

  #Cosmology

  2013

  The number densities of protons of mass $m_{p}$ or neutrons of mass $m_{n}$ in kinetic equilibrium at temperature $T$, in the absence of any chemical potentials, are each given by (with $i=n$ or $p$ )

  $$n_{i}=g_{i}\left(\frac{m_{i} k_{B} T}{2 \pi \hbar^{2}}\right)^{3 / 2} \exp \left[-m_{i} c^{2} / k_{B} T\right]$$

  where $k_{B}$ is Boltzmann's constant and $g_{i}$ is the spin degeneracy.

  Use this to show, to a very good approximation, that the ratio of the number of neutrons to protons at a temperature $T \simeq 1 \mathrm{MeV} / k_{B}$ is given by

  $$\frac{n_{n}}{n_{p}}=\exp \left[-\left(m_{n}-m_{p}\right) c^{2} / k_{B} T\right]$$

  where $\left(m_{n}-m_{p}\right) c^{2}=1.3 \mathrm{MeV}$. Explain any approximations you have used.

  The reaction rate for weak interactions between protons and neutrons at energies $5 \mathrm{MeV} \geqslant k_{B} T \geqslant 0.8 \mathrm{MeV}$ is given by $\Gamma=\left(k_{B} T / 1 \mathrm{MeV}\right)^{5} \mathrm{~s}^{-1}$ and the expansion rate of the universe at these energies is given by $H=\left(k_{B} T / 1 M e V\right)^{2} s^{-1}$. Give an example of a weak interaction that can maintain equilibrium abundances of protons and neutrons at these energies. Show how the final abundance of neutrons relative to protons can be calculated and use it to estimate the mass fraction of the universe in helium- 4 after nucleosynthesis.

  What would have happened to the helium abundance if the proton and neutron masses had been exactly equal?

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• Paper 2, Section I, D

  #Cosmology

  2013

  The linearised equation for the growth of small inhomogeneous density perturbations $\delta_{\mathbf{k}}$ with comoving wavevector $\mathbf{k}$ in an isotropic and homogeneous universe is

  $$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}+\left(\frac{c_{s}^{2} k^{2}}{a^{2}}-4 \pi G \rho\right) \delta_{\mathbf{k}}=0$$

  where $\rho$ is the matter density, $c_{s}=(d P / d \rho)^{1 / 2}$ is the sound speed, $P$ is the pressure, $a(t)$ is the expansion scale factor of the unperturbed universe, and overdots denote differentiation with respect to time $t$.

  Define the Jeans wavenumber and explain its physical meaning.

  Assume the unperturbed Friedmann universe has zero curvature and cosmological constant and it contains only zero-pressure matter, so that $a(t)=a_{0} t^{2 / 3}$. Show that the solution for the growth of density perturbations is given by

  $$\delta_{\mathbf{k}}=A(\mathbf{k}) t^{2 / 3}+B(\mathbf{k}) t^{-1}$$

  Comment briefly on the cosmological significance of this result.

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• Paper 1, Section I, D

  #Cosmology

  2013

  The Friedmann equation and the fluid conservation equation for a closed isotropic and homogeneous cosmology are given by

  $$\begin{aligned} &\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{1}{a^{2}} \\ &\dot{\rho}+3 \frac{\dot{a}}{a}(\rho+P)=0 \end{aligned}$$

  where the speed of light is set equal to unity, $G$ is the gravitational constant, $a(t)$ is the expansion scale factor, $\rho$ is the fluid mass density and $P$ is the fluid pressure, and overdots denote differentiation with respect to the time coordinate $t$.

  If the universe contains only blackbody radiation and $a=0$ defines the zero of time $t$, show that

  $$a^{2}(t)=t\left(t_{*}-t\right)$$

  where $t_{*}$ is a constant. What is the physical significance of the time $t_{*}$ ? What is the value of the ratio $a(t) / t$ at the time when the scale factor is largest? Sketch the curve of $a(t)$ and identify its geometric shape.

  Briefly comment on whether this cosmological model is a good description of the observed universe at any time in its history.

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• Paper 3, Section II, D

  #Cosmology

  2013

  The contents of a spatially homogeneous and isotropic universe are modelled as a finite mass $M$ of pressureless material whose radius $r(t)$ evolves from some constant reference radius $r_{0}$ in proportion to the time-dependent scale factor $a(t)$, with

  $$r(t)=a(t) r_{0}$$

  (i) Show that this motion leads to expansion governed by Hubble's Law. If this universe is expanding, explain why there will be a shift in the frequency of radiation between its emission from a distant object and subsequent reception by an observer. Define the redshift $z$ of the observed object in terms of the values of the scale factor $a(t)$ at the times of emission and reception.

  (ii) The expanding universal mass $M$ is given a small rotational perturbation, with angular velocity $\omega$, and its angular momentum is subsequently conserved. If deviations from spherical expansion can be neglected, show that its linear rotational velocity will fall as $V \propto a^{-n}$, where you should determine the value of $n$. Show that this perturbation will become increasingly insignificant compared to the expansion velocity as the universe expands if $a \propto t^{2 / 3}$.

  (iii) A distant cloud of intermingled hydrogen (H) atoms and carbon monoxide (CO) molecules has its redshift determined simultaneously in two ways: by detecting $21 \mathrm{~cm}$ radiation from atomic hydrogen and by detecting radiation from rotational transitions in CO molecules. The ratio of the $21 \mathrm{~cm}$ atomic transition frequency to the CO rotational transition frequency is proportional to $\alpha^{2}$, where $\alpha$ is the fine structure constant. It is suggested that there may be a small difference in the value of the constant $\alpha$ between the times of emission and reception of the radiation from the cloud.

  Show that the difference in the redshift values for the cloud, $\Delta z=z_{C O}-z_{21}$, determined separately by observations of the $\mathrm{H}$ and $\mathrm{CO}$ transitions, is related to $\delta \alpha=$ $\alpha_{r}-\alpha_{e}$, the difference in $\alpha$ values at the times of reception and emission, by

  $$\Delta z=2\left(\frac{\delta \alpha}{\alpha_{r}}\right)\left(1+z_{C O}\right)$$

  (iv) The universe today contains $30 \%$ of its total density in the form of pressureless matter and $70 \%$ in the form of a dark energy with constant redshift-independent density. If these are the only two significant constituents of the universe, show that their densities were equal when the scale factor of the universe was approximately equal to $75 \%$ of its present value.

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• Paper 1, Section II, D

  #Cosmology

  2013

  A spherically symmetric star of total mass $M_{s}$ has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the radial distance from its centre. These quantities are related by the equations of hydrostatic equilibrium and mass conservation:

  $$\begin{aligned} \frac{d P}{d r} &=-\frac{G M(r) \rho}{r^{2}} \\ \frac{d M}{d r} &=4 \pi \rho r^{2} \end{aligned}$$

  where $M(r)$ is the mass inside radius $r$.

  By integrating from the centre of the star at $r=0$, where $P=P_{c}$, to the surface of the star at $r=R_{s}$, where $P=P_{s}$, show that

  $$4 \pi R_{s}^{3} P_{s}=\Omega+3 \int_{0}^{M_{s}} \frac{P}{\rho} d M,$$

  where $\Omega$ is the total gravitational potential energy. Show that

  $$-\Omega>\frac{G M_{s}^{2}}{2 R_{s}}$$

  If the surface pressure is negligible and the star is a perfect gas of particles of mass $m$ with number density $n$ and $P=n k_{B} T$ at temperature $T$, and radiation pressure can be ignored, then show that

  $$3 \int_{0}^{M_{s}} \frac{P}{\rho} d M=\frac{3 k_{B}}{m} \bar{T},$$

  where $\bar{T}$ is the mean temperature of the star, which you should define.

  Hence, show that the mean temperature of the star satisfies the inequality

  $$\bar{T}>\frac{G M_{s} m}{6 k_{B} R_{s}}$$

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• Paper 4, Section II, H

  #Differential Geometry

  2013

  Define what is meant by the geodesic curvature $k_{g}$ of a regular curve $\alpha: I \rightarrow S$ parametrized by arc length on a smooth oriented surface $S \subset \mathbf{R}^{3}$. If $S$ is the unit sphere in $\mathbf{R}^{3}$ and $\alpha: I \rightarrow S$ is a parametrized geodesic circle of radius $\phi$, with $0<\phi<\pi / 2$, justify the fact that $\left|k_{g}\right|=\cot \phi$.

  State the general form of the Gauss-Bonnet theorem with boundary on an oriented surface $S$, explaining briefly the terms which occur.

  Let $S \subset \mathbf{R}^{3}$ now denote the circular cone given by $z>0$ and $x^{2}+y^{2}=z^{2} \tan ^{2} \phi$, for a fixed choice of $\phi$ with $0<\phi<\pi / 2$, and with a fixed choice of orientation. Let $\alpha: I \rightarrow S$ be a simple closed piecewise regular curve on $S$, with (signed) exterior angles $\theta_{1}, \ldots, \theta_{N}$ at the vertices (that is, $\theta_{i}$ is the angle between limits of tangent directions, with sign determined by the orientation). Suppose furthermore that the smooth segments of $\alpha$ are geodesic curves. What possible values can $\theta_{1}+\cdots+\theta_{N}$ take? Justify your answer.

  [You may assume that a simple closed curve in $\mathbf{R}^{2}$ bounds a region which is homeomorphic to a disc. Given another simple closed curve in the interior of this region, you may assume that the two curves bound a region which is homeomorphic to an annulus.]

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• Paper 3, Section II, H

  #Differential Geometry

  2013

  We say that a parametrization $\phi: U \rightarrow S \subset \mathbf{R}^{3}$ of a smooth surface $S$ is isothermal if the coefficients of the first fundamental form satisfy $F=0$ and $E=G=\lambda(u, v)^{2}$, for some smooth non-vanishing function $\lambda$ on $U$. For an isothermal parametrization, prove that

  $$\phi_{u u}+\phi_{v v}=2 \lambda^{2} H \mathbf{N}$$

  where $\mathbf{N}$ denotes the unit normal vector and $H$ the mean curvature, which you may assume is given by the formula

  $$H=\frac{g+e}{2 \lambda^{2}}$$

  where $g=-\left\langle\mathbf{N}_{u}, \phi_{u}\right\rangle$ and $e=-\left\langle\mathbf{N}_{v}, \phi_{v}\right\rangle$ are coefficients in the second fundamental form.

  Given a parametrization $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ of a surface $S \subset \mathbf{R}^{3}$, we consider the complex valued functions on $U$ :

  $$\theta_{1}=x_{u}-i x_{v}, \quad \theta_{2}=y_{u}-i y_{v}, \quad \theta_{3}=z_{u}-i z_{v}$$

  Show that $\phi$ is isothermal if and only if $\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}=0$. If $\phi$ is isothermal, show that $S$ is a minimal surface if and only if $\theta_{1}, \theta_{2}, \theta_{3}$ are holomorphic functions of the complex variable $\zeta=u+i v$

  Consider the holomorphic functions on $D:=\mathbf{C} \backslash \mathbf{R}_{\geqslant 0}$ (with complex coordinate $\zeta=u+i v$ on $\mathbf{C})$ given by

  $$\theta_{1}:=\frac{1}{2}\left(1-\zeta^{-2}\right), \quad \theta_{2}:=-\frac{i}{2}\left(1+\zeta^{-2}\right), \quad \theta_{3}:=-\zeta^{-1}$$

  Find a smooth map $\phi(u, v)=(x(u, v), y(u, v), z(u, v)): D \rightarrow \mathbf{R}^{3}$ for which $\phi(-1,0)=\mathbf{0}$ and the $\theta_{i}$ defined by (2) satisfy the equations (1). Show furthermore that $\phi$ extends to a smooth map $\tilde{\phi}: \mathbf{C}^{*} \rightarrow \mathbf{R}^{3}$. If $w=x+i y$ is the complex coordinate on $\mathbf{C}$, show that

  $$\widetilde{\phi}(\exp (i w))=(\cosh y \cos x+1, \cosh y \sin x, y)$$

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• Paper 2, Section II, H

  #Differential Geometry

  2013

  Let $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ be a regular curve parametrized by arc length having nowherevanishing curvature. State the Frenet relations between the tangent, normal and binormal vectors at a point, and their derivatives.

  Let $S \subset \mathbf{R}^{3}$ be a smooth oriented surface. Define the Gauss map $N: S \rightarrow S^{2}$, and show that its derivative at $P \in S, d N_{P}: T_{P} S \rightarrow T_{P} S$, is self-adjoint. Define the Gaussian curvature of $S$ at $P$.

  Now suppose that $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ has image in $S$ and that its normal curvature is zero for all $s \in[0, L]$. Show that the Gaussian curvature of $S$ at a point $P=\alpha(s)$ of the curve is $K(P)=-\tau(s)^{2}$, where $\tau(s)$ denotes the torsion of the curve.

  If $S \subset \mathbf{R}^{3}$ is a standard embedded torus, show that there is a curve on $S$ for which the normal curvature vanishes and the Gaussian curvature of $S$ is zero at all points of the curve.

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• Paper 1, Section II, H

  #Differential Geometry

  2013

  For $f: X \rightarrow Y$ a smooth map of manifolds, define the concepts of critical point, critical value and regular value.

  With the obvious identification of $\mathbf{C}$ with $\mathbf{R}^{2}$, and hence also of $\mathbf{C}^{3}$ with $\mathbf{R}^{6}$, show that the complex-valued polynomial $z_{1}^{3}+z_{2}^{2}+z_{3}^{2}$ determines a smooth map $f: \mathbf{R}^{6} \rightarrow \mathbf{R}^{2}$ whose only critical point is at the origin. Hence deduce that $V:=f^{-1}((0,0)) \backslash\{\mathbf{0}\} \subset \mathbf{R}^{6}$ is a 4-dimensional manifold, and find the equations of its tangent space at any given point $\left(z_{1}, z_{2}, z_{3}\right) \in V$.

  Now let $S^{5} \subset \mathbf{C}^{3}=\mathbf{R}^{6}$ be the unit 5 -sphere, defined by $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}=1$. Given a point $P=\left(z_{1}, z_{2}, z_{3}\right) \in S^{5} \cap V$, by considering the vector $\left(2 z_{1}, 3 z_{2}, 3 z_{3}\right) \in \mathbf{C}^{3}=\mathbf{R}^{6}$ or otherwise, show that not all tangent vectors to $V$ at $P$ are tangent to $S^{5}$. Deduce that $S^{5} \cap V \subset \mathbf{R}^{6}$ is a compact three-dimensional manifold.

  [Standard results may be quoted without proof if stated carefully.]

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• Paper 4, Section I, C

  #Dynamical Systems

  2013

  Consider the system

  $$\begin{aligned} &\dot{x}=y+a x+b x^{3} \\ &\dot{y}=-x \end{aligned}$$

  What is the Poincaré index of the single fixed point? If there is a closed orbit, why must it enclose the origin?

  By writing $\dot{x}=\partial H / \partial y+g(x)$ and $\dot{y}=-\partial H / \partial x$ for suitable functions $H(x, y)$ and $g(x)$, show that if there is a closed orbit $\mathcal{C}$ then

  $$\oint_{\mathcal{C}}\left(a x+b x^{3}\right) x d t=0$$

  Deduce that there is no closed orbit when $a b>0$.

  If $a b<0$ and $a$ and $b$ are both $O(\epsilon)$, where $\epsilon$ is a small parameter, then there is a single closed orbit that is to within $O(\epsilon)$ a circle of radius $R$ centred on the origin. Deduce a relation between $a, b$ and $R$.

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• Paper 3, Section I, C

  #Dynamical Systems

  2013

  A one-dimensional map is defined by

  $$x_{n+1}=F\left(x_{n}, \mu\right)$$

  where $\mu$ is a parameter. What is the condition for a bifurcation of a fixed point $x_{*}$ of $F$ ?

  Let $F(x, \mu)=x\left(x^{2}-2 x+\mu\right)$. Find the fixed points and show that bifurcations occur when $\mu=-1, \mu=1$ and $\mu=2$. Sketch the bifurcation diagram, showing the locus and stability of the fixed points in the $(x, \mu)$ plane and indicating the type of each bifurcation.

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• Paper 2, Section I, $7 \mathrm{C}$

  #Dynamical Systems

  2013

  Let $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ be a two-dimensional dynamical system with a fixed point at $\mathbf{x}=\mathbf{0}$. Define a Lyapunov function $V(\mathbf{x})$ and explain what it means for $\mathbf{x}=\mathbf{0}$ to be Lyapunov stable.

  For the system

  $$\begin{aligned} &\dot{x}=-x-2 y+x^{3} \\ &\dot{y}=-y+x+\frac{1}{2} y^{3}+x^{2} y \end{aligned}$$

  determine the values of $C$ for which $V=x^{2}+C y^{2}$ is a Lyapunov function in a sufficiently small neighbourhood of the origin.

  For the case $C=2$, find $V_{1}$ and $V_{2}$ such that $V(\mathbf{x})<V_{1}$ at $t=0$ implies that $V \rightarrow 0$ as $t \rightarrow \infty$ and $V(\mathbf{x})>V_{2}$ at $t=0$ implies that $V \rightarrow \infty$ as $t \rightarrow \infty$

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• Paper 1, Section I, C

  #Dynamical Systems

  2013

  Consider the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$ which has a hyperbolic fixed point at the origin.

  Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.

  Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?

  For the system

  $$\begin{aligned} &\dot{x}=-x+x^{2}+y^{2} \\ &\dot{y}=y+y^{2}-x^{2} \end{aligned}$$

  calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.

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• Paper 3, Section II, C

  #Dynamical Systems

  2013

  Let $f: I \rightarrow I$ be a continuous map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (a) $f$ has a horseshoe and (b) $f$ is chaotic according to Glendinning's definition of chaos.

  Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$, $x_{0}<x_{1}<x_{2}$. Prove that $f^{2}$ has a horseshoe. [You may assume the Intermediate Value Theorem.]

  Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph. Explain how the existence of the 3 -cycle corresponds to this graph.

  The map $g: I \rightarrow I$ has a 4-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}\right\}$ with $x_{1}=g\left(x_{0}\right), x_{2}=g\left(x_{1}\right)$, $x_{3}=g\left(x_{2}\right)$ and $x_{0}=g\left(x_{3}\right)$. If $x_{0}<x_{3}<x_{2}<x_{1}$ is $g$ necessarily chaotic? [You may use a suitable directed graph as part of your argument.]

  How does your answer change if $x_{0}<x_{2}<x_{1}<x_{3}$ ?

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• Paper 4, Section II, C

  #Dynamical Systems

  2013

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=(x+y+a)(x-y+a) \\ &\dot{y}=y-x^{2}-b \end{aligned}$$

  where $a>0$.

  Find the fixed points of the dynamical system. Show that for any fixed value of $a$ there exist three values $b_{1}>b_{2} \geqslant b_{3}$ of $b$ where a bifurcation occurs. Show that $b_{2}=b_{3}$ when $a=1 / 2$.

  In the remainder of this question set $a=1 / 2$.

  (i) Being careful to explain your reasoning, show that the extended centre manifold for the bifurcation at $b=b_{1}$ can be written in the form $X=\alpha Y+\beta \mu+p(Y, \mu)$, where $X$ and $Y$ denote the departures from the values of $x$ and $y$ at the fixed point, $b=b_{1}+\mu, \alpha$ and $\beta$ are suitable constants (to be determined) and $p$ is quadratic to leading order. Derive a suitable approximate form for $p$, and deduce the nature of the bifurcation and the stability of the different branches of the steady state solution near the bifurcation.

  (ii) Repeat the calculations of part (i) for the bifurcation at $b=b_{2}$.

  (iii) Sketch the $x$ values of the fixed points as functions of $b$, indicating the nature of the bifurcations and where each branch is stable.

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• Paper 4, Section II, 35B

  #Electrodynamics

  2013

  (i) For a time-dependent source, confined within a domain $D$, show that the time derivative $\dot{d}$ of the dipole moment $\mathbf{d}$ satisfies

  $$\dot{\mathbf{d}}=\int_{D} d^{3} y \mathbf{J}(\mathbf{y})$$

  where $\mathbf{J}$ is the current density.

  (ii) The vector potential $\mathbf{A}(\mathbf{x}, t)$ due to a time-dependent source is given by

  $$\mathbf{A}=\frac{1}{r} f(t-r / c) \mathbf{k}$$

  where $r=|\mathbf{x}| \neq 0$, and $\mathbf{k}$ is the unit vector in the $z$ direction. Calculate the resulting magnetic field $\mathbf{B}(\mathbf{x}, t)$. By considering the magnetic field for small $r$ show that the dipole moment of the effective source satisfies

  $$\frac{\mu_{0}}{4 \pi} \dot{\mathbf{d}}=f(t) \mathbf{k}$$

  Calculate the asymptotic form of the magnetic field $\mathbf{B}$ at very large $r$.

  (iii) Using the equation

  $$\frac{\partial \mathbf{E}}{\partial t}=c^{2} \nabla \times \mathbf{B}$$

  calculate $\mathbf{E}$ at very large $r$. Show that $\mathbf{E}, \mathbf{B}$ and $\hat{\mathbf{r}}=\mathbf{x} /|\mathbf{x}|$ form a right-handed triad, and moreover $|\mathbf{E}|=c|\mathbf{B}|$. How do $|\mathbf{E}|$ and $|\mathbf{B}|$ depend on $r ?$ What is the significance of this?

  (iv) Calculate the power $P(\theta, \phi)$ emitted per unit solid angle and sketch its dependence on $\theta$. Show that the emitted radiation is polarised and describe how the plane of polarisation (that is, the plane in which $\mathbf{E}$ and $\hat{\mathbf{r}}$ lie) depends on the direction of the dipole. Suppose the dipole moment has constant amplitude and constant frequency and so the radiation is monochromatic with wavelength $\lambda$. How does the emitted power depend on $\lambda$ ?

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• Paper 3, Section II, 36B

  #Electrodynamics

  2013

  (i) Obtain Maxwell's equations in empty space from the action functional

  $$S\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$

  where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$.

  (ii) A modification of Maxwell's equations has the action functional

  $$\tilde{S}\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x\left\{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}+\frac{1}{2 \lambda^{2}} A_{\mu} A^{\mu}\right\},$$

  where again $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and $\lambda$ is a constant. Obtain the equations of motion of this theory and show that they imply $\partial_{\mu} A^{\mu}=0$.

  (iii) Show that the equations of motion derived from $\tilde{S}$ admit solutions of the form

  $$A^{\mu}=A_{0}^{\mu} e^{i k_{\nu} x^{\nu}}$$

  where $A_{0}^{\mu}$ is a constant 4-vector, and the 4 -vector $k_{\mu}$ satisfies $A_{0}^{\mu} k_{\mu}=0$ and $k_{\mu} k^{\mu}=-1 / \lambda^{2}$.

  (iv) Show further that the tensor

  $$T_{\mu \nu}=\frac{1}{\mu_{0}}\left\{F_{\mu \sigma} F_{\nu}^{\sigma}-\frac{1}{4} \eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta}-\frac{1}{2 \lambda^{2}}\left(\eta_{\mu \nu} A_{\alpha} A^{\alpha}-2 A_{\mu} A_{\nu}\right)\right\}$$

  is conserved, that is $\partial^{\mu} T_{\mu \nu}=0$.

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• Paper 1, Section II, 36B

  #Electrodynamics

  2013

  (i) Starting from

  $$F^{\mu \nu}=\left(\begin{array}{cccc} 0 & E_{1} / c & E_{2} / c & E_{3} / c \\ -E_{1} / c & 0 & B_{3} & -B_{2} \\ -E_{2} / c & -B_{3} & 0 & B_{1} \\ -E_{3} / c & B_{2} & -B_{1} & 0 \end{array}\right)$$

  and performing a Lorentz transformation with $\gamma=1 / \sqrt{1-u^{2} / c^{2}}$, using

  $$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma u / c & 0 & 0 \\ -\gamma u / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  show how $\mathbf{E}$ and $\mathbf{B}$ transform under a Lorentz transformation.

  (ii) By taking the limit $c \rightarrow \infty$, obtain the behaviour of $\mathbf{E}$ and $\mathbf{B}$ under a Galilei transfomation and verify the invariance under Galilei transformations of the nonrelativistic equation

  $$m \frac{d \mathbf{v}}{d t}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})$$

  (iii) Show that Maxwell's equations admit solutions of the form

  $$\mathbf{E}=\mathbf{E}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c), \quad \mathbf{B}=\mathbf{B}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c)$$

  where $f$ is an arbitrary function, $\mathbf{n}$ is a unit vector, and the constant vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are subject to restrictions which should be stated.

  (iv) Perform a Galilei transformation of a solution $(\star)$, with $\mathbf{n}=(1,0,0)$. Show that, by a particular choice of $u$, the solution may brought to the form

  $$\tilde{\mathbf{E}}=\tilde{\mathbf{E}}_{0} g(\tilde{x}), \quad \tilde{\mathbf{B}}=\tilde{\mathbf{B}}_{0} g(\tilde{x}),$$

  where $g$ is an arbitrary function and $\tilde{x}$ is a spatial coordinate in the rest frame. By showing that $(t)$ is not a solution of Maxwell's equations in the boosted frame, conclude that Maxwell's equations are not invariant under Galilei transformations.

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• Paper 4, Section II, A

  #Fluid Dynamics II

  2013

  Consider the flow of an incompressible fluid of uniform density $\rho$ and dynamic viscosity $\mu$. Show that the rate of viscous dissipation per unit volume is given by

  $$\Phi=2 \mu e_{i j} e_{i j}$$

  where $e_{i j}$ is the strain rate.

  Determine expressions for $e_{i j}$ and $\Phi$ when the flow is irrotational with velocity potential $\phi$.

  In deep water a linearised wave with a surface displacement $\eta=a \cos (k x-\omega t)$ has a velocity potential $\phi=-(\omega a / k) e^{-k z} \sin (k x-\omega t)$. Hence determine the rate of the viscous dissipation, averaged over a wave period $2 \pi / \omega$, for an irrotational surface wave of wavenumber $k$ and small amplitude $a \ll 1 / k$ in a fluid with very small viscosity $\mu \ll \rho \omega / k^{2}$ and great depth $H \gg 1 / k$.

  Calculate the depth-integrated kinetic energy per unit wavelength. Assuming that the average potential energy is equal to the average kinetic energy, show that the total wave energy decreases to leading order as $e^{-\gamma t}$, where $\gamma$ should be found.

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• Paper 2, Section II, A

  #Fluid Dynamics II

  2013

  Write down the boundary-layer equations for steady two-dimensional flow of a viscous incompressible fluid with velocity $U(x)$ outside the boundary layer. Find the boundary layer thickness $\delta(x)$ when $U(x)=U_{0}$, a constant. Show that the boundarylayer equations can be satisfied in this case by a streamfunction $\psi(x, y)=g(x) f(\eta)$ with suitable scaling function $g(x)$ and similarity variable $\eta$. Find the equation satisfied by $f$ and the associated boundary conditions.

  Find the drag on a thin two-dimensional flat plate of finite length $L$ placed parallel to a uniform flow. Why does the drag not increase in proportion to the length of the plate? [You may assume that the boundary-layer solution is applicable except in negligibly small regions near the leading and trailing edges. You may also assume that $f^{\prime \prime}(0)=0.33$.]

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• Paper 3, Section II, A

  #Fluid Dynamics II

  2013

  A disk hovers on a cushion of air above an air-table - a fine porous plate through which a constant flux of air is pumped. Let the disk have a radius $R$ and a weight $M g$ and hover at a low height $h \ll R$ above the air-table. Let the volume flux of air, which has density $\rho$ and viscosity $\mu$, be $w$ per unit surface area. The conditions are such that $\rho w h^{2} / \mu R \ll 1$. Explain the significance of this restriction.

  Find the pressure distribution in the air under the disk. Show that this pressure balances the weight of the disk if

  $$h=R\left(\frac{3 \pi \mu R w}{2 M g}\right)^{1 / 3}$$

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• Paper 1, Section II, A

  #Fluid Dynamics II

  2013

  The velocity field $\mathbf{u}$ and stress tensor $\sigma$ satisfy the Stokes equations in a volume $V$ bounded by a surface $S$. Let $\hat{\mathbf{u}}$ be another solenoidal velocity field. Show that

  $$\int_{S} \sigma_{i j} n_{j} \hat{u}_{i} d S=\int_{V} 2 \mu e_{i j} \hat{e}_{i j} d V$$

  where $\mathbf{e}$ and $\hat{\mathbf{e}}$ are the strain-rates corresponding to the velocity fields $\mathbf{u}$ and $\hat{\mathbf{u}}$ respectively, and $\mathbf{n}$ is the unit normal vector out of $V$. Hence, or otherwise, prove the minimum dissipation theorem for Stokes flow.

  A particle moves at velocity $\mathbf{U}$ through a highly viscous fluid of viscosity $\mu$ contained in a stationary vessel. As the particle moves, the fluid exerts a drag force $\mathbf{F}$ on it. Show that

  $$-\mathbf{F} \cdot \mathbf{U}=\int_{V} 2 \mu e_{i j} e_{i j} d V .$$

  Consider now the case when the particle is a small cube, with sides of length $\ell$, moving in a very large vessel. You may assume that

  $$\mathbf{F}=-k \mu \ell \mathbf{U}$$

  for some constant $k$. Use the minimum dissipation theorem, being careful to declare the domain(s) involved, to show that

  $$3 \pi \leqslant k \leqslant 3 \sqrt{3} \pi .$$

  [You may assume Stokes' result for the drag on a sphere of radius $a, \mathbf{F}=-6 \pi \mu a \mathbf{U}$.]

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• Paper 4, Section I, E

  #Further Complex Methods

  2013

  Let the function $f(z)$ be analytic in the upper half-plane and such that $|f(z)| \rightarrow 0$ as $|z| \rightarrow \infty$. Show that

  $$\mathcal{P} \int_{-\infty}^{\infty} \frac{f(x)}{x} d x=i \pi f(0),$$

  where $\mathcal{P}$ denotes the Cauchy principal value.

  Use the Cauchy integral theorem to show that

  $$\mathcal{P} \int_{-\infty}^{\infty} \frac{u(x, 0)}{x-t} d x=-\pi v(t, 0), \quad \mathcal{P} \int_{-\infty}^{\infty} \frac{v(x, 0)}{x-t} d x=\pi u(t, 0),$$

  where $u(x, y)$ and $v(x, y)$ are the real and imaginary parts of $f(z)$.

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• Paper 3, Section I, E

  #Further Complex Methods

  2013

  Let a real-valued function $u=u(x, y)$ be the real part of a complex-valued function $f=f(z)$ which is analytic in the neighbourhood of a point $z=0$, where $z=x+i y .$ Derive a formula for $f$ in terms of $u$ and use it to find an analytic function $f$ whose real part is

  $$\frac{x^{3}+x^{2}-y^{2}+x y^{2}}{(x+1)^{2}+y^{2}}$$

  and such that $f(0)=0$.

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• Paper 2, Section I, E

  #Further Complex Methods

  2013

  (i) Find all branch points of $\left(z^{3}-1\right)^{1 / 4}$ on an extended complex plane.

  (ii) Use a branch cut to evaluate the integral

  $$\int_{-2}^{2}\left(4-x^{2}\right)^{1 / 2} d x$$

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• Paper 1, Section I, E

  #Further Complex Methods

  2013

  Prove that there are no second order linear ordinary homogeneous differential equations for which all points in the extended complex plane are analytic.

  Find all such equations which have one regular singular point at $z=0$.

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• Paper 2, Section II, E

  #Further Complex Methods

  2013

  The Beta function is defined for $\operatorname{Re}(z)>0$ as

  $$B(z, q)=\int_{0}^{1} t^{q-1}(1-t)^{z-1} d t, \quad(\operatorname{Re}(q)>0)$$

  and by analytic continuation elsewhere in the complex $z$-plane.

  Show that:

  (i) $(z+q) B(z+1, q)=z B(z, q)$;

  (ii) $\Gamma(z)^{2}=B(z, z) \Gamma(2 z)$.

  By considering $\Gamma\left(z / 2^{m}\right)$ for all positive integers $m$, deduce that $\Gamma(z) \neq 0$ for all $z$ with $\operatorname{Re}(z)>0$.

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• Paper 1, Section II, E

  #Further Complex Methods

  2013

  Show that the equation

  $$(z-1) w^{\prime \prime}-z w^{\prime}+(4-2 z) w=0$$

  has solutions of the form $w(z)=\int_{\gamma} \exp (z t) f(t) d t$, where

  $$f(t)=\frac{\exp (-t)}{(t-a)(t-b)^{2}}$$

  and the contour $\gamma$ is any closed curve in the complex plane, where $a$ and $b$ are real constants which should be determined.

  Use this to find the general solution, evaluating the integrals explicitly.

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• Paper 4, Section II, I

  #Galois Theory

  2013

  (i) Let $\zeta_{N}=\mathrm{e}^{2 \pi i / N} \in \mathbb{C}$ for $N \geqslant 1$. For the cases $N=11,13$, is it possible to express $\zeta_{N}$, starting with integers and using rational functions and (possibly nested) radicals? If it is possible, briefly explain how this is done, assuming standard facts in Galois Theory.

  (ii) Let $F=\mathbb{C}(X, Y, Z)$ be the rational function field in three variables over $\mathbb{C}$, and for integers $a, b, c \geqslant 1$ let $K=\mathbb{C}\left(X^{a}, Y^{b}, Z^{c}\right)$ be the subfield of $F$ consisting of all rational functions in $X^{a}, Y^{b}, Z^{c}$ with coefficients in $\mathbb{C}$. Show that $F / K$ is Galois, and determine its Galois group. [Hint: For $\alpha, \beta, \gamma \in \mathbb{C}^{\times}$, the map $(X, Y, Z) \longmapsto(\alpha X, \beta Y, \gamma Z)$ is an automorphism of $F$.]

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• Paper 3, Section II, I

  #Galois Theory

  2013

  Let $p$ be a prime number and $F$ a field of characteristic $p$. Let $\operatorname{Fr}_{p}: F \rightarrow F$ be the Frobenius map defined by $\operatorname{Fr}_{p}(x)=x^{p}$ for all $x \in F$.

  (i) Prove that $\operatorname{Fr}_{p}$ is a field automorphism when $F$ is a finite field.

  (ii) Is the same true for an arbitrary algebraic extension $F$ of $\mathbb{F}_{p}$ ? Justify your answer.

  (iii) Let $F=\mathbb{F}_{p}\left(X_{1}, \ldots, X_{n}\right)$ be the rational function field in $n$ variables where $n \geqslant 1$ over $\mathbb{F}_{p}$. Determine the image of $\operatorname{Fr}_{p}: F \rightarrow F$, and show that $\operatorname{Fr}_{p}$ makes $F$ into an extension of degree $p^{n}$ over a subfield isomorphic to $F$. Is it a separable extension?

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• Paper 2, Section II, I

  #Galois Theory

  2013

  For a positive integer $N$, let $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ be the cyclotomic field obtained by adjoining all $N$-th roots of unity to $\mathbb{Q}$. Let $F=\mathbb{Q}\left(\boldsymbol{\mu}_{24}\right)$.

  (i) Determine the Galois group of $F$ over $\mathbb{Q}$.

  (ii) Find all $N>1$ such that $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ is contained in $F$.

  (iii) List all quadratic and quartic extensions of $\mathbb{Q}$ which are contained in $F$, in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Indicate which of these fields occurred in (ii).

  [Standard facts on the Galois groups of cyclotomic fields and the fundamental theorem of Galois theory may be used freely without proof.]

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• Paper 1, Section II, I

  #Galois Theory

  2013

  (i) Give an example of a field $F$, contained in $\mathbb{C}$, such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $F[X]$. Justify your answer.

  (ii) Let $F$ be any extension of degree 3 over $\mathbb{Q}$. Prove that the polynomial $X^{4}+1$ is irreducible over $F$.

  (iii) Give an example of a prime number $p$ such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $\mathbb{F}_{p}[X]$. Justify your answer.

  [Standard facts on fields, extensions, and finite fields may be quoted without proof, as long as they are stated clearly.]

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• Paper 4, Section II, D

  #General Relativity

  2013

  Consider the metric describing the interior of a star,

  $$d s^{2}=-e^{2 \alpha(r)} d t^{2}+e^{2 \beta(r)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  defined for $0 \leqslant r \leqslant r_{0}$ by

  $$e^{\alpha(r)}=\frac{3}{2} e^{-\beta_{0}}-\frac{1}{2} e^{-\beta(r)},$$

  with

  $$e^{-2 \beta(r)}=1-A r^{2}$$

  Here $A=2 M / r_{0}^{3}$, where $M$ is the mass of the star, $\beta_{0}=\beta\left(r_{0}\right)$, and we have taken units in which we have set $G=c=1$.

  (i) The star is made of a perfect fluid with energy-momentum tensor

  $$T_{a b}=(p+\rho) u_{a} u_{b}+p g_{a b}$$

  Here $u^{a}$ is the 4-velocity of the fluid which is at rest, the density $\rho$ is constant throughout the star $\left(0 \leqslant r \leqslant r_{0}\right)$ and the pressure $p=p(r)$ depends only on the radial coordinate. Write down the Einstein field equations and show that they may be written as

  $$R_{a b}=8 \pi(p+\rho) u_{a} u_{b}+4 \pi(\rho-p) g_{a b}$$

  (ii) Using the formulae given below, or otherwise, show that for $0 \leqslant r \leqslant r_{0}$, one has

  $$\begin{aligned} &4 \pi(\rho+p)=\frac{\left(\alpha^{\prime}+\beta^{\prime}\right)}{r} e^{-2 \beta(r)} \\ &4 \pi(\rho-p)=\left(\frac{\beta^{\prime}-\alpha^{\prime}}{r}-\frac{1}{r^{2}}\right) e^{-2 \beta(r)}+\frac{1}{r^{2}} \end{aligned}$$

  where primes denote differentiation with respect to $r$. Hence show that

  $$\rho=\frac{3 A}{8 \pi} \quad, \quad p(r)=\frac{3 A}{8 \pi}\left(\frac{e^{-\beta(r)}-e^{-\beta_{0}}}{3 e^{-\beta_{0}}-e^{-\beta(r)}}\right)$$

  [The non-zero components of the Ricci tensor are

  $$\begin{aligned} &R_{00}=e^{2 \alpha-2 \beta}\left(\alpha^{\prime \prime}-\alpha^{\prime} \beta^{\prime}+\alpha^{\prime 2}+\frac{2 \alpha^{\prime}}{r}\right) \\ &R_{11}=-\alpha^{\prime \prime}+\alpha^{\prime} \beta^{\prime}-\alpha^{\prime 2}+\frac{2 \beta^{\prime}}{r} \\ &R_{22}=1+e^{-2 \beta}\left[\left(\beta^{\prime}-\alpha^{\prime}\right) r-1\right] \\ &R_{33}=\sin ^{2} \theta R_{22} \end{aligned}$$

  Note that

  $$\left.\alpha^{\prime}=\frac{1}{2} A r e^{\beta-\alpha} \quad, \quad \beta^{\prime}=A r e^{2 \beta} .\right]$$

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• Paper 2, Section II, D

  #General Relativity

  2013

  A spacetime contains a one-parameter family of geodesics $x^{a}=x^{a}(\lambda, \mu)$, where $\lambda$ is a parameter along each geodesic, and $\mu$ labels the geodesics. The tangent to the geodesics is $T^{a}=\partial x^{a} / \partial \lambda$, and $N^{a}=\partial x^{a} / \partial \mu$ is a connecting vector. Prove that

  $$\nabla_{\mu} T^{a}=\nabla_{\lambda} N^{a}$$

  and hence derive the equation of geodesic deviation:

  $$\nabla_{\lambda}^{2} N^{a}+R_{b c d}^{a} T^{b} N^{c} T^{d}=0$$

  [You may assume $R_{b c d}^{a}=-R_{b d c}^{a}$ and the Ricci identity in the form

  $$\left.\left(\nabla_{\lambda} \nabla_{\mu}-\nabla_{\mu} \nabla_{\lambda}\right) T^{a}=R_{b c d}^{a} T^{b} T^{c} N^{d}\right]$$

  Consider the two-dimensional space consisting of the sphere of radius $r$ with line element

  $$d s^{2}=r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) .$$

  Show that one may choose $T^{a}=(1,0), N^{a}=(0,1)$, and that

  $$\nabla_{\theta} N^{a}=\cot \theta N^{a} .$$

  Hence show that $R=2 / r^{2}$, using the geodesic deviation equation and the identity in any two-dimensional space

  $$R_{b c d}^{a}=\frac{1}{2} R\left(\delta_{c}^{a} g_{b d}-\delta_{d}^{a} g_{b c}\right)$$

  where $R$ is the Ricci scalar.

  Verify your answer by direct computation of $R$.

  [You may assume that the only non-zero connection components are

  $$\Gamma_{\phi \theta}^{\phi}=\Gamma_{\theta \phi}^{\phi}=\cot \theta$$

  and

  $$\Gamma_{\phi \phi}^{\theta}=-\sin \theta \cos \theta .$$

  You may also use the definition

  $$\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{e c}^{a} \Gamma_{b d}^{e}-\Gamma_{e d}^{a} \Gamma_{b c}^{e}\right]$$

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• Paper 3, Section II, D

  #General Relativity

  2013

  The Schwarzschild metric for a spherically symmetric black hole is given by

  $$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  where we have taken units in which we set $G=c=1$. Consider a photon moving within the equatorial plane $\theta=\frac{\pi}{2}$, along a path $x^{a}(\lambda)$ with affine parameter $\lambda$. Using a variational principle with Lagrangian

  $$L=g_{a b} \frac{d x^{a}}{d \lambda} \frac{d x^{b}}{d \lambda}$$

  or otherwise, show that

  $$\left(1-\frac{2 M}{r}\right)\left(\frac{d t}{d \lambda}\right)=E \quad \text { and } \quad r^{2}\left(\frac{d \phi}{d \lambda}\right)=h$$

  for constants $E$ and $h$. Deduce that

  $$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

  Assume now that the photon approaches from infinity. Show that the impact parameter (distance of closest approach) is given by

  $$b=\frac{h}{E}$$

  Denote the right hand side of equation $(*)$ as $f(r)$. By sketching $f(r)$ in each of the cases below, or otherwise, show that:

  (a) if $b^{2}>27 M^{2}$, the photon is deflected but not captured by the black hole;

  (b) if $b^{2}<27 M^{2}$, the photon is captured;

  (c) if $b^{2}=27 M^{2}$, the photon orbit has a particular form, which should be described.

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• Paper 1, Section II, 37D

  #General Relativity

  2013

  The curve $\gamma, x^{a}=x^{a}(\lambda)$, is a geodesic with affine parameter $\lambda$. Write down the geodesic equation satisfied by $x^{a}(\lambda)$.

  Suppose the parameter is changed to $\mu(\lambda)$, where $d \mu / d \lambda>0$. Obtain the corresponding equation and find the condition for $\mu$ to be affine. Deduce that, whatever parametrization $\nu$ is used along the curve $\gamma$, the tangent vector $K^{a}$ to $\gamma$ satisfies

  $$\left(\nabla_{\nu} K\right)^{[a} K^{b]}=0 \text {. }$$

  Now consider a spacetime with metric $g_{a b}$, and conformal transformation

  $$\tilde{g}_{a b}=\Omega^{2}\left(x^{c}\right) g_{a b}$$

  The curve $\gamma$ is a geodesic of the metric connection of $g_{a b}$. What further restriction has to be placed on $\gamma$ so that it is also a geodesic of the metric connection of $\tilde{g}_{a b}$ ? Justify your answer.

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• Paper 4, Section I, $3 G$

  #Geometry and Groups

  2013

  Let $\Delta_{1}, \Delta_{2}$ be two disjoint closed discs in the Riemann sphere with bounding circles $\Gamma_{1}, \Gamma_{2}$ respectively. Let $J_{k}$ be inversion in the circle $\Gamma_{k}$ and let $T$ be the Möbius transformation $J_{2} \circ J_{1}$.

  Show that, if $w \notin \Delta_{1}$, then $T(w) \in \Delta_{2}$ and so $T^{n}(w) \in \Delta_{2}$ for $n=1,2,3, \ldots$ Deduce that $T$ has a fixed point in $\Delta_{2}$ and a second in $\Delta_{1}$.

  Deduce that there is a Möbius transformation $A$ with

  $$A\left(\Delta_{1}\right)=\{z:|z| \leqslant 1\} \quad \text { and } \quad A\left(\Delta_{2}\right)=\{z:|z| \geqslant R\}$$

  for some $R>1$.

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• Paper 3, Section I, $3 G$

  #Geometry and Groups

  2013

  Let $\Lambda$ be a rank 2 lattice in the Euclidean plane. Show that the group $G$ of all Euclidean isometries of the plane that map $\Lambda$ onto itself is a discrete group. List the possible sizes of the point groups for $G$ and give examples to show that point groups of these sizes do arise.

  [You may quote any standard results without proof.]

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• Paper 2, Section I, G

  #Geometry and Groups

  2013

  Let $\ell_{1}, \ell_{2}$ be two straight lines in Euclidean 3-space. Show that there is a rotation about some axis through an angle $\pi$ that maps $\ell_{1}$ onto $\ell_{2}$. Is this rotation unique?

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• Paper 1, Section I, G

  #Geometry and Groups

  2013

  Show that any pair of lines in hyperbolic 3-space that does not have a common endpoint must have a common normal. Is this still true when the pair of lines does have a common endpoint?

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• Paper 1, Section II, G

  #Geometry and Groups

  2013

  Define the modular group $\Gamma$ acting on the upper half-plane.

  Describe the set $S$ of points $z$ in the upper half-plane that have $\operatorname{Im}(T(z)) \leqslant \operatorname{Im}(z)$ for each $T \in \Gamma$. Hence find a fundamental set for $\Gamma$ acting on the upper half-plane.

  Let $A$ and $J$ be the two Möbius transformations

  $$A: z \mapsto z+1 \quad \text { and } \quad J: z \mapsto-1 / z$$

  When is $\operatorname{Im}(J(z))>\operatorname{Im}(z) ?$

  For any point $z$ in the upper half-plane, show that either $z \in S$ or else there is an integer $k$ with

  $$\operatorname{Im}\left(J\left(A^{k}(z)\right)\right)>\operatorname{Im}(z)$$

  Deduce that the modular group is generated by $A$ and $J$.

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• Paper 4, Section II, $12 \mathrm{G}$

  #Geometry and Groups

  2013

  Define the limit set for a Kleinian group. If your definition of the limit set requires an arbitrary choice of a base point, you should prove that the limit set does not depend on this choice.

  Let $\Delta_{1}, \Delta_{2}, \Delta_{3}, \Delta_{4}$ be the four discs $\{z \in \mathbb{C}:|z-c| \leqslant 1\}$ where $c$ is the point $1+i, 1-i,-1-i,-1+i$ respectively. Show that there is a parabolic Möbius transformation $A$ that maps the interior of $\Delta_{1}$ onto the exterior of $\Delta_{2}$ and fixes the point where $\Delta_{1}$ and $\Delta_{2}$ touch. Show further that we can choose $A$ so that it maps the unit disc onto itself.

  Let $B$ be the similar parabolic transformation that maps the interior of $\Delta_{3}$ onto the exterior of $\Delta_{4}$, fixes the point where $\Delta_{3}$ and $\Delta_{4}$ touch, and maps the unit disc onto itself. Explain why the group generated by $A$ and $B$ is a Kleinian group $G$. Find the limit set for the group $G$ and justify your answer.

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• Paper 4, Section II, F

  #Graph Theory

  2013

  Define the maximum degree $\Delta(G)$ and the chromatic index $\chi^{\prime}(G)$ of the graph $G$.

  State and prove Vizing's theorem relating $\Delta(G)$ and $\chi^{\prime}(G)$.

  Let $G$ be a connected graph such that $\chi^{\prime}(G)=\Delta(G)+1$ but, for every subgraph $H$ of $G, \chi^{\prime}(H)=\Delta(H)$ holds. Show that $G$ is a circuit of odd length.

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• Paper 3, Section II, F

  #Graph Theory

  2013

  Let $G$ be a graph of order $n$ and average degree $d$. Let $A$ be the adjacency matrix of $G$ and let $x^{n}+c_{1} x^{n-1}+c_{2} x^{n-2}+\cdots+c_{n}$ be its characteristic polynomial. Show that $c_{1}=0$ and $c_{2}=-n d / 2$. Show also that $-c_{3}$ is twice the number of triangles in $G$.

  The eigenvalues of $A$ are $\lambda_{1} \geqslant \lambda_{2} \geqslant \cdots \geqslant \lambda_{n}$. Prove that $\lambda_{1} \geqslant d$.

  Evaluate $\lambda_{1}+\cdots+\lambda_{n}$. Show that $\lambda_{1}^{2}+\cdots+\lambda_{n}^{2}=n d$ and infer that $\lambda_{1} \leqslant \sqrt{d(n-1)}$. Does there exist, for each $n$, a graph $G$ with $d>0$ for which $\lambda_{2}=\cdots=\lambda_{n}$ ?

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• Paper 2, Section II, F

  #Graph Theory

  2013

  Let $G$ be a graph with $|G| \geqslant 3$. State and prove a necessary and sufficient condition for $G$ to be Eulerian (that is, for $G$ to have an Eulerian circuit).

  Prove that if $\delta(G) \geqslant|G| / 2$ then $G$ is Hamiltonian (that is, $G$ has a Hamiltonian circuit).

  The line graph $L(G)$ of $G$ has vertex set $V(L(G))=E(G)$ and edge set

  $$E(L(G))=\{e f: e, f \in E(G), e \text { and } f \text { are incident }\} .$$

  Show that $L(G)$ is Eulerian if $G$ is regular and connected.

  Must $L(G)$ be Hamiltonian if $G$ is Eulerian? Must $G$ be Eulerian if $L(G)$ is Hamiltonian? Justify your answers.

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• Paper 1, Section II, F

  #Graph Theory

  2013

  State and prove Hall's theorem about matchings in bipartite graphs.

  Show that a regular bipartite graph has a matching meeting every vertex.

  A graph is almost r-regular if each vertex has degree $r-1$ or $r$. Show that, if $r \geqslant 2$, an almost $r$-regular graph $G$ must contain an almost $(r-1)$-regular graph $H$ with $V(H)=V(G)$.

  [Hint: First, if possible, remove edges from $G$ whilst keeping it almost r-regular.]

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• Paper 3, Section II, C

  #Integrable Systems

  2013

  Let $U=U(x, y)$ and $V=V(x, y)$ be two $n \times n$ complex-valued matrix functions, smoothly differentiable in their variables. We wish to explore the solution of the overdetermined linear system

  $$\frac{\partial \mathbf{v}}{\partial y}=U(x, y) \mathbf{v}, \quad \frac{\partial \mathbf{v}}{\partial x}=V(x, y) \mathbf{v}$$

  for some twice smoothly differentiable vector function $\mathbf{v}(x, y)$.

  Prove that, if the overdetermined system holds, then the functions $U$ and $V$ obey the zero curvature representation

  $$\frac{\partial U}{\partial x}-\frac{\partial V}{\partial y}+U V-V U=0$$

  Let $u=u(x, y)$ and

  $$U=\left[\begin{array}{cc} i \lambda & i \bar{u} \\ i u & -i \lambda \end{array}\right], \quad V=\left[\begin{array}{cc} 2 i \lambda^{2}-i|u|^{2} & 2 i \lambda \bar{u}+\bar{u}_{y} \\ 2 i \lambda u-u_{y} & -2 i \lambda^{2}+i|u|^{2} \end{array}\right]$$

  where subscripts denote derivatives, $\bar{u}$ is the complex conjugate of $u$ and $\lambda$ is a constant. Find the compatibility condition on the function $u$ so that $U$ and $V$ obey the zero curvature representation.

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• Paper 2, Section II, 32C

  #Integrable Systems

  2013

  Consider the Hamiltonian system

  $$\mathbf{p}^{\prime}=-\frac{\partial H}{\partial \mathbf{q}}, \quad \mathbf{q}^{\prime}=\frac{\partial H}{\partial \mathbf{p}}$$

  where $H=H(\mathbf{p}, \mathbf{q})$.

  When is the transformation $\mathbf{P}=\mathbf{P}(\mathbf{p}, \mathbf{q}), \mathbf{Q}=\mathbf{Q}(\mathbf{p}, \mathbf{q})$ canonical?

  Prove that, if the transformation is canonical, then the equations in the new variables $(\mathbf{P}, \mathbf{Q})$ are also Hamiltonian, with the same Hamiltonian function $H$.

  Let $\mathbf{P}=C^{-1} \mathbf{p}+B q, \mathbf{Q}=C \mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation to be canonical.

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• Paper 1, Section II, C

  #Integrable Systems

  2013

  Quoting carefully all necessary results, use the theory of inverse scattering to derive the 1-soliton solution of the $\mathrm{KdV}$ equation

  $$u_{t}=6 u u_{x}-u_{x x x}$$

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• Paper 3, Section II, F

  #Linear Analysis

  2013

  State the Stone-Weierstrass Theorem for real-valued functions.

  State Riesz's Lemma.

  Let $K$ be a compact, Hausdorff space and let $A$ be a subalgebra of $C(K)$ separating the points of $K$ and containing the constant functions. Fix two disjoint, non-empty, closed subsets $E$ and $F$ of $K$.

  (i) If $x \in E$ show that there exists $g \in A$ such that $g(x)=0,0 \leqslant g<1$ on $K$, and $g>0$ on $F$. Explain briefly why there is $M \in \mathbb{N}$ such that $g \geqslant \frac{2}{M}$ on $F$.

  (ii) Show that there is an open cover $U_{1}, U_{2}, \ldots, U_{m}$ of $E$, elements $g_{1}, g_{2}, \ldots, g_{m}$ of $A$ and positive integers $M_{1}, M_{2}, \ldots, M_{m}$ such that

  $$0 \leqslant g_{r}<1 \text { on } K, \quad g_{r} \geqslant \frac{2}{M_{r}} \text { on } F, \quad g_{r}<\frac{1}{2 M_{r}} \text { on } U_{r}$$

  for each $r=1,2, \ldots, m$.

  (iii) Using the inequality

  $$1-N t \leqslant(1-t)^{N} \leqslant \frac{1}{N t} \quad(0<t<1, N \in \mathbb{N})$$

  show that for sufficiently large positive integers $n_{1}, n_{2}, \ldots, n_{m}$, the element

  $$h_{r}=1-\left(1-g_{r}^{n_{r}}\right)^{M_{r}^{n_{r}}}$$

  of $A$ satisfies

  $$0 \leqslant h_{r} \leqslant 1 \text { on } K, \quad h_{r} \leqslant \frac{1}{4} \text { on } U_{r}, \quad h_{r} \geqslant\left(\frac{3}{4}\right)^{\frac{1}{m}} \text { on } F$$

  for each $r=1,2, \ldots, m$.

  (iv) Show that the element $h=h_{1} \cdot h_{2} \cdots \cdot h_{m}-\frac{1}{2}$ of $A$ satisfies

  $$-\frac{1}{2} \leqslant h \leqslant \frac{1}{2} \text { on } K, \quad h \leqslant-\frac{1}{4} \text { on } E, \quad h \geqslant \frac{1}{4} \text { on } F \text {. }$$

  Now let $f \in C(K)$ with $\|f\| \leqslant 1$. By considering the sets $\left\{x \in K: f(x) \leqslant-\frac{1}{4}\right\}$ and $\left\{x \in K: f(x) \geqslant \frac{1}{4}\right\}$, show that there exists $h \in A$ such that $\|f-h\| \leqslant \frac{3}{4}$. Deduce that $A$ is dense in $C(K)$.

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• Paper 4, Section II, F

  #Linear Analysis

  2013

  Let $T: X \rightarrow X$ be a bounded linear operator on a complex Banach space $X$. Define the spectrum $\sigma(T)$ of $T$. What is an approximate eigenvalue of $T$ ? What does it mean to say that $T$ is compact?

  Assume now that $T$ is compact. Show that if $\lambda$ is in the boundary of $\sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$. [You may use without proof the result that every $\lambda$ in the boundary of $\sigma(T)$ is an approximate eigenvalue of $T$.]

  Let $T: H \rightarrow H$ be a compact Hermitian operator on a complex Hilbert space $H$. Prove the following:

  (a) If $\lambda \in \sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$.

  (b) $\sigma(T)$ is countable.

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• Paper 2, Section II, F

  #Linear Analysis

  2013

  Let $X$ be a Banach space. Let $T: X \rightarrow \ell_{\infty}$ be a bounded linear operator. Show that there is a bounded sequence $\left(f_{n}\right)_{n=1}^{\infty}$ in $X^{*}$ such that $T x=\left(f_{n} x\right)_{n=1}^{\infty}$ for all $x \in X$.

  Fix $1<p<\infty$. Define the Banach space $\ell_{p}$ and briefly explain why it is separable. Show that for $x \in \ell_{p}$ there exists $f \in \ell_{p}^{*}$ such that $\|f\|=1$ and $f(x)=\|x\|_{p}$. [You may use Hölder's inequality without proof.]

  Deduce that $\ell_{p}$ embeds isometrically into $\ell_{\infty}$.

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• Paper 1, Section II, F

  #Linear Analysis

  2013

  State and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]

  Let $X$ be a closed subspace of $\ell_{\infty}$ such that $X$ is also a subset of $\ell_{1}$. Show that the left-shift $L: X \rightarrow \ell_{1}$, given by $L\left(x_{1}, x_{2}, x_{3}, \ldots\right)=\left(x_{2}, x_{3}, \ldots\right)$, is bounded when $X$ is equipped with the sup-norm.

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• Paper 2, Section II, G

  #Logic and Set Theory

  2013

  Explain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem for such posets.

  A poset $P$ is called directed if every finite subset of $P$ (including the empty subset) has an upper bound in $P ; P$ is called directed-complete if every subset of $P$ which is directed (in the induced ordering) has a least upper bound in $P$. Show that the set of all chains in an arbitrary poset $P$, ordered by inclusion, is directed-complete.

  Given a poset $P$, let $[P \rightarrow P]$ denote the set of all order-preserving maps $P \rightarrow P$, ordered pointwise (i.e. $f \leqslant g$ if and only if $f(x) \leqslant g(x)$ for all $x$ ). Show that $[P \rightarrow P]$ is directed-complete if $P$ is.

  Now suppose $P$ is directed-complete, and that $f: P \rightarrow P$ is order-preserving and inflationary. Show that there is a unique smallest set $C \subseteq[P \rightarrow P]$ satisfying

  (a) $f \in C$;

  (b) $C$ is closed under composition (i.e. $g, h \in C \Rightarrow g \circ h \in C$ ); and

  (c) $C$ is closed under joins of directed subsets.

  Show that

  (i) all maps in $C$ are inflationary;

  (ii) $C$ is directed;

  (iii) if $g=\bigvee C$, then all values of $g$ are fixed points of $f$;

  (iv) for every $x \in P$, there exists $y \in P$ with $x \leqslant y=f(y)$.

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• Paper 3, Section II, G

  #Logic and Set Theory

  2013

  Explain carefully what is meant by syntactic entailment and semantic entailment in the propositional calculus. State the Completeness Theorem for the propositional calculus, and deduce the Compactness Theorem.

  Suppose $P, Q$ and $R$ are pairwise disjoint sets of primitive propositions, and let $\phi \in \mathcal{L}(P \cup Q)$ and $\psi \in \mathcal{L}(Q \cup R)$ be propositional formulae such that $(\phi \Rightarrow \psi)$ is a theorem of the propositional calculus. Consider the set

  $$X=\{\chi \in \mathcal{L}(Q) \mid \phi \vdash \chi\}$$

  Show that $X \cup\{\neg \psi\}$ is inconsistent, and deduce that there exists $\chi \in \mathcal{L}(Q)$ such that both $(\phi \Rightarrow \chi)$ and $(\chi \Rightarrow \psi)$ are theorems. [Hint: assuming $X \cup\{\neg \psi\}$ is consistent, take a suitable valuation $v$ of $Q \cup R$ and show that

  $$\{\phi\} \cup\{q \in Q \mid v(q)=1\} \cup\{\neg q \mid q \in Q, v(q)=0\}$$

  is inconsistent. The Deduction Theorem may be assumed without proof.]

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• Paper 4, Section II, G

  #Logic and Set Theory

  2013

  State the Axiom of Foundation and the Principle of $\in$-Induction, and show that they are equivalent in the presence of the other axioms of ZF set theory. [You may assume the existence of transitive closures.]

  Given a model $(V, \in)$ for all the axioms of ZF except Foundation, show how to define a transitive class $R$ which, with the restriction of the given relation $\in$, is a model of ZF.

  Given a model $(V, \in)$ of $\mathrm{ZF}$, indicate briefly how one may modify the relation $\in$ so that the resulting structure $\left(V, \in^{\prime}\right)$ fails to satisfy Foundation, but satisfies all the other axioms of $\mathrm{ZF}$. [You need not verify that all the other axioms hold in $\left(V, \in^{\prime}\right)$.]

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• Paper 1, Section II, G

  #Logic and Set Theory

  2013

  Write down the recursive definitions of ordinal addition, multiplication and exponentiation.

  Given that $F: \mathbf{O n} \rightarrow \mathbf{O n}$ is a strictly increasing function-class (i.e. $\alpha<\beta$ implies $F(\alpha)<F(\beta))$, show that $\alpha \leqslant F(\alpha)$ for all $\alpha$.

  Show that every ordinal $\alpha$ has a unique representation in the form

  $$\alpha=\omega^{\alpha_{1}} \cdot a_{1}+\omega^{\alpha_{2}} \cdot a_{2}+\cdots+\omega^{\alpha_{n}} \cdot a_{n}$$

  where $n \in \omega, \alpha \geqslant \alpha_{1}>\alpha_{2}>\cdots>\alpha_{n}$, and $a_{1}, a_{2}, \ldots, a_{n} \in \omega \backslash\{0\}$.

  Under what conditions can an ordinal $\alpha$ be represented in the form

  $$\omega^{\beta_{1}} \cdot b_{1}+\omega^{\beta_{2}} \cdot b_{2}+\cdots+\omega^{\beta_{m}} \cdot b_{m}$$

  where $\beta_{1}<\beta_{2}<\cdots<\beta_{m}$ and $b_{1}, b_{2}, \ldots, b_{m} \in \omega \backslash\{0\} ?$ Justify your answer.

  [The laws of ordinal arithmetic (associative, distributive, etc.) may be assumed without proof.]

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• Paper 4, Section I, A

  #Mathematical Biology

  2013

  A model of two populations competing for resources takes the form

  $$\begin{aligned} \frac{d n_{1}}{d t} &=r_{1} n_{1}\left(1-n_{1}-a_{12} n_{2}\right) \\ \frac{d n_{2}}{d t} &=r_{2} n_{2}\left(1-n_{2}-a_{21} n_{1}\right) \end{aligned}$$

  where all parameters are positive. Give a brief biological interpretation of $a_{12}, a_{21}, r_{1}$ and $r_{2}$. Briefly describe the dynamics of each population in the absence of the other.

  Give conditions for there to exist a steady-state solution with both populations present (that is, $n_{1}>0$ and $n_{2}>0$ ), and give conditions for this solution to be stable.

  In the case where there exists a solution with both populations present but the solution is not stable, what is the likely long-term outcome for the biological system? Explain your answer with the aid of a phase diagram in the $\left(n_{1}, n_{2}\right)$ plane.

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• Paper 3, Section I, A

  #Mathematical Biology

  2013

  An immune system creates a burst of $N$ new white blood cells with probability $b$ per unit time. White blood cells die with probability $d$ each per unit time. Write down the master equation for $P_{n}(t)$, the probability that there are $n$ white blood cells at time $t$.

  Given that $n=n_{0}$ initially, find an expression for the mean of $n$.

  Show that the variance of $n$ has the form $A e^{-2 d t}+B e^{-d t}+C$ and find $A, B$ and $C$.

  If the immune system were modified to produce $k$ times as many cells per burst but with probability per unit time divided by a factor $k$, how would the mean and variance of the number of cells change?

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• Paper 2, Section I, A

  #Mathematical Biology

  2013

  The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies

  $$\frac{\partial n(a, t)}{\partial t}+\frac{\partial n(a, t)}{\partial a}=-\mu(a) n(a, t)$$

  with

  $$n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

  where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age

  Seek a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ and show that

  $$r(a)=r(0) e^{-\gamma a-\int_{0}^{a} \mu(s) d s}, \quad r(0)=\int_{0}^{\infty} b(s) r(s) d s .$$

  Show also that if

  $$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a=1$$

  then there is such a similarity solution. Give a biological interpretation of $\phi(0)$.

  Suppose now that all births happen at age $a^{*}$, at which time an individual produces $B$ offspring, and that the death rate is constant with age (i.e. $\mu(a)=\mu)$. Find the similarity solution and give the condition for this to represent a growing population.

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• Paper 1, Section I, A

  #Mathematical Biology

  2013

  In a discrete-time model, a proportion $\mu$ of mature bacteria divides at each time step. When a mature bacterium divides it is destroyed and two new immature bacteria are produced. A proportion $\lambda$ of the immature bacteria matures at each time step, and a proportion $k$ of mature bacteria dies at each time step. Show that this model may be represented by the equations

  $$\begin{aligned} &a_{t+1}=a_{t}+2 \mu b_{t}-\lambda a_{t} \\ &b_{t+1}=b_{t}-\mu b_{t}+\lambda a_{t}-k b_{t} \end{aligned}$$

  Give an expression for the general solution to these equations and show that the population may grow if $\mu>k$.

  At time $T$, the population is treated with an antibiotic that completely stops bacteria from maturing, but otherwise has no direct effects. Explain what will happen to the population of bacteria afterwards, and give expressions for $a_{t}$ and $b_{t}$ for $t>T$ in terms of $a_{T}, b_{T}, \mu$ and $k$.

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• Paper 3, Section II, A

  #Mathematical Biology

  2013

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}+u-u v+a u^{2} \\ &\frac{\partial v}{\partial t}=d \frac{\partial^{2} v}{\partial x^{2}}+u^{2}-b u v \end{aligned}$$

  where $a, b, d>0$.

  Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

  Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(d, b)$ plane for fixed $a \in(0,1)$.

  Find $k_{c}$, the critical wavenumber at the onset of the instability, in terms of $a$ and $b$.

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• Paper 2, Section II, A

  #Mathematical Biology

  2013

  The concentration $c(x, t)$ of insects at position $x$ at time $t$ satisfies the nonlinear diffusion equation

  $$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{m} \frac{\partial c}{\partial x}\right)$$

  with $m>0$. Find the value of $\alpha$ which allows a similarity solution of the form $c(x, t)=t^{\alpha} f(\xi)$, with $\xi=t^{\alpha} x$.

  Show that

  $$f(\xi)=\left\{\begin{array}{cl} {\left[\frac{\alpha m}{2}\left(\xi^{2}-\xi_{0}^{2}\right)\right]^{1 / m}} & \text { for }-\xi_{0}<\xi<\xi_{0} \\ 0 & \text { otherwise } \end{array}\right.$$

  where $\xi_{0}$ is a constant. From the original partial differential equation, show that the total number of insects $c_{0}$ does not change in time. From this result, find a general expression relating $\xi_{0}$ and $c_{0}$. Find a closed-form solution for $\xi_{0}$ in the case $m=2$.

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• Paper 4, Section II, H

  #Number Fields

  2013

  State Dedekind's criterion. Use it to factor the primes up to 5 in the ring of integers $\mathcal{O}_{K}$ of $K=\mathbb{Q}(\sqrt{65})$. Show that every ideal in $\mathcal{O}_{K}$ of norm 10 is principal, and compute the class group of $K$.

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• Paper 2, Section II, H

  #Number Fields

  2013

  (i) State Dirichlet's unit theorem.

  (ii) Let $K$ be a number field. Show that if every conjugate of $\alpha \in \mathcal{O}_{K}$ has absolute value at most 1 then $\alpha$ is either zero or a root of unity.

  (iii) Let $k=\mathbb{Q}(\sqrt{3})$ and $K=\mathbb{Q}(\zeta)$ where $\zeta=e^{i \pi / 6}=(i+\sqrt{3}) / 2$. Compute $N_{K / k}(1+\zeta)$. Show that

  $$\mathcal{O}_{K}^{*}=\left\{(1+\zeta)^{m} u: 0 \leqslant m \leqslant 11, u \in \mathcal{O}_{k}^{*}\right\}$$

  Hence or otherwise find fundamental units for $k$ and $K$.

  [You may assume that the only roots of unity in $K$ are powers of $\zeta .$ ]

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• Paper 1, Section II, H

  #Number Fields

  2013

  Let $f \in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f$.

  (i) Show that if $\operatorname{disc}(f)$ is square-free then $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

  (ii) In the case $f(X)=X^{3}-3 X-25$ find the minimal polynomial of $\beta=3 /(1-\alpha)$ and hence compute the discriminant of $K$. What is the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ ?

  [Recall that the discriminant of $X^{3}+p X+q$ is $-4 p^{3}-27 q^{2}$.]

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• Paper 1, Section I, I

  #Number Theory

  2013

  State and prove Gauss's Lemma for the Legendre symbol $\left(\frac{a}{p}\right)$. For which odd primes $p$ is 2 a quadratic residue modulo $p ?$ Justify your answer.

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• Paper 4, Section I, I

  #Number Theory

  2013

  Let $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$. Define the Riemann zeta function $\zeta(s)$ for $\sigma>1$. Show that for $\sigma>1$,

  $$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$$

  where the product is taken over all primes. Deduce that there are infinitely many primes.

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• Paper 3, Section I, I

  #Number Theory

  2013

  State the Chinese Remainder Theorem.

  A composite number $n$ is defined to be a Carmichael number if $b^{n-1} \equiv 1 \bmod n$ whenever $(b, n)=1$. Show that a composite $n$ is Carmichael if and only if $n$ is square-free and $(p-1)$ divides $(n-1)$ for all prime factors $p$ of $n$. [You may assume that, for $p$ an odd prime and $\alpha \geqslant 1$ an integer, $\left(\mathbb{Z} / p^{\alpha} \mathbb{Z}\right)^{\times}$is a cyclic group.]

  Show that if $n=(6 t+1)(12 t+1)(18 t+1)$ with all three factors prime, then $n$ is Carmichael.

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• Paper 2, Section I, I

  #Number Theory

  2013

  Define Euler's totient function $\phi(n)$, and show that $\sum_{d \mid n} \phi(d)=n$. Hence or otherwise prove that for any prime $p$ the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.

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• Paper 4, Section II, I

  #Number Theory

  2013

  (i) What is meant by the continued fraction expansion of a real number $\theta$ ? Suppose that $\theta$ has continued fraction $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ to $\theta$ and give the recurrence relations satisfied by the $p_{n}$ and $q_{n}$. Show that the convergents $p_{n} / q_{n}$ do indeed converge to $\theta$.

  [You need not justify the basic order properties of finite continued fractions.]

  (ii) Find two solutions in strictly positive integers to each of the equations

  $$x^{2}-10 y^{2}=1 \quad \text { and } \quad x^{2}-11 y^{2}=1$$

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• Paper 3, Section II, I

  #Number Theory

  2013

  Define equivalence of binary quadratic forms and show that equivalent forms have the same discriminant.

  Show that an integer $n$ is properly represented by a binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d \bmod 4 n$ is soluble in integers. Which primes are represented by a form of discriminant $-20$ ?

  What does it mean for a positive definite form to be reduced? Find all reduced forms of discriminant $-20$. For each member of your list find the primes less than 100 represented by the form.

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• Paper 4, Section II, C

  #Numerical Analysis

  2013

  Consider the solution of the two-point boundary value problem

  $$(2-\sin \pi x) u^{\prime \prime}+u=1, \quad-1 \leqslant x \leqslant 1$$

  with periodic boundary conditions at $x=-1$ and $x=1$. Construct explicitly the linear algebraic system that arises from the application of a spectral method to the above equation.

  The Fourier coefficients of $u$ are defined by

  $$\hat{u}_{n}=\frac{1}{2} \int_{-1}^{1} u(\tau) e^{-i \pi n \tau} d \tau$$

  Prove that the computation of the Fourier coefficients for the truncated system with $-N / 2+1 \leqslant n \leqslant N / 2$ (where $N$ is an even and positive integer, and assuming that $\hat{u}_{n}=0$ outside this range of $n$ ) reduces to the solution of a tridiagonal system of algebraic equations, which you should specify.

  Explain the term convergence with spectral speed and justify its validity for the derived approximation of $u$.

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• Paper 2, Section II, C

  #Numerical Analysis

  2013

  Consider the advection equation $u_{t}=u_{x}$ on the unit interval $x \in[0,1]$ and $t \geqslant 0$, where $u=u(x, t)$, subject to the initial condition $u(x, 0)=\varphi(x)$ and the boundary condition $u(1, t)=0$, where $\varphi$ is a given smooth function on $[0,1]$.

  (i) We commence by discretising the advection equation above with finite differences on the equidistant space-time grid $\{(m \Delta x, n \Delta t), m=0, \ldots, M+1, n=0, \ldots, T\}$ with $\Delta x=1 /(M+1)$ and $\Delta t>0$. We obtain an equation for $u_{m}^{n} \approx u(m \Delta x, n \Delta t)$ that reads

  $$u_{m}^{n+1}=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right), \quad m=1, \ldots, M, n \in \mathbb{Z}^{+}$$

  with the condition $u_{0}^{n}=0$ for all $n \in \mathbb{Z}^{+}$and $\mu=\Delta t / \Delta x$.

  What is the order of approximation (that is, the order of the local error) in space and time of the above discrete solution to the exact solution of the advection equation? Write the scheme in matrix form and deduce for which choices of $\mu$ this approximation converges to the exact solution. State (without proof) any theorems you use. [You may use the fact that for a tridiagonal $M \times M$ matrix

  $$\left(\begin{array}{cccc} \alpha & \beta & 0 & 0 \\ -\beta & \ddots & \ddots & 0 \\ 0 & \ddots & \ddots & \beta \\ 0 & 0 & -\beta & \alpha \end{array}\right)$$

  the eigenvalues are given by $\lambda_{\ell}=\alpha+2 i \beta \cos \frac{\ell \pi}{M+1}$.]

  (ii) How does the order change when we replace the central difference approximation of the first derivative in space by forward differences, that is $u_{m+1}^{n}-u_{m}^{n}$ instead of $\left(u_{m+1}^{n}-u_{m-1}^{n}\right) / 2 ?$ For which choices of $\mu$ is this new scheme convergent?

  (iii) Instead of the approximation in (i) we consider the following method for numerically solving the advection equation,

  $$u_{m}^{n+1}=\mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right)+u_{m}^{n-1}$$

  where we additionally assume that $u_{m}^{1}$ is given. What is the order of this method for a fixed $\mu$ ?

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• Paper 3, Section II, C

  #Numerical Analysis

  2013

  (i) Suppose that $A$ is a real $n \times n$ matrix, and that $\mathbf{w} \in \mathbb{R}^{n}$ and $\lambda_{1} \in \mathbb{R}$ are given so that $A \mathbf{w}=\lambda_{1} \mathbf{w}$. Further, let $S$ be a non-singular matrix such that $S \mathbf{w}=c \mathbf{e}_{1}$, where $\mathbf{e}_{1}$ is the first coordinate vector and $c \neq 0$. Let $\widehat{A}=S A S^{-1}$. Prove that the eigenvalues of $A$ are $\lambda_{1}$ together with the eigenvalues of the bottom right $(n-1) \times(n-1)$ submatrix of $\widehat{A}$.

  (ii) Suppose again that $A$ is a real $n \times n$ matrix, and that two linearly independent vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^{n}$ are given such that the linear subspace $\mathcal{L}\{\mathbf{v}, \mathbf{w}\}$ spanned by $\mathbf{v}$ and $w$ is invariant under the action of $A$, that is

  $$x \in \mathcal{L}\{\mathbf{v}, \mathbf{w}\} \quad \Rightarrow \quad A x \in \mathcal{L}\{\mathbf{v}, \mathbf{w}\} .$$

  Denote by $V$ an $n \times 2$ matrix whose two columns are the vectors $\mathbf{v}$ and $\mathbf{w}$, and let $S$ be a non-singular matrix such that $R=S V$ is upper triangular, that is

  $$R=S V=S \times\left(\begin{array}{cc} v_{1} & w_{1} \\ v_{2} & w_{2} \\ \vdots & \vdots \\ v_{n} & w_{n} \end{array}\right)=\left(\begin{array}{cc} r_{11} & r_{12} \\ 0 & r_{22} \\ 0 & 0 \\ \vdots & \vdots \\ 0 & 0 \end{array}\right)$$

  Again, let $\widehat{A}=S A S^{-1}$. Prove that the eigenvalues of $A$ are the eigenvalues of the top left $2 \times 2$ submatrix of $\widehat{A}$ together with the eigenvalues of the bottom right $(n-2) \times(n-2)$ submatrix of $\widehat{A} .$

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• Paper 1, Section II, 40C

  #Numerical Analysis

  2013

  Let

  $$A(\alpha)=\left(\begin{array}{ccc} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \end{array}\right), \quad \alpha \in \mathbb{R}$$

  (i) For which values of $\alpha$ is $A(\alpha)$ positive definite?

  (ii) Formulate the Gauss-Seidel method for the solution $\mathbf{x} \in \mathbb{R}^{3}$ of a system

  $$A(\alpha) \mathbf{x}=\mathbf{b}$$

  with $A(\alpha)$ as defined above and $\mathbf{b} \in \mathbb{R}^{3}$. Prove that the Gauss-Seidel method converges to the solution of the above system whenever $A$ is positive definite. [You may state and use the Householder-John theorem without proof.]

  (iii) For which values of $\alpha$ does the Jacobi iteration applied to the solution of the above system converge?

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• Paper 4, Section II, K

  #Optimization and Control

  2013

  Given $r, \rho, \mu, T$, all positive, it is desired to choose $u(t)>0$ to maximize

  $$\mu x(T)+\int_{0}^{T} e^{-\rho t} \log u(t) d t$$

  subject to $\dot{x}(t)=r x(t)-u(t), x(0)=10$.

  Explain what Pontryagin's maximum principle guarantees about a solution to this problem.

  Show that no matter whether $x(T)$ is constrained or unconstrained there is a constant $\alpha$ such that the optimal control is of the form $u(t)=\alpha e^{-(\rho-r) t}$. Find an expression for $\alpha$ under the constraint $x(T)=5$.

  Show that if $x(T)$ is unconstrained then $\alpha=(1 / \mu) e^{-r T}$.

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• Paper 3, Section II, K

  #Optimization and Control

  2013

  A particle follows a discrete-time trajectory in $\mathbb{R}^{2}$ given by

  $$\left(\begin{array}{l} x_{t+1} \\ y_{t+1} \end{array}\right)=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)\left(\begin{array}{l} x_{t} \\ y_{t} \end{array}\right)+\left(\begin{array}{c} t \\ 1 \end{array}\right) u_{t}+\left(\begin{array}{c} \epsilon_{t} \\ 0 \end{array}\right)$$

  where $\left\{\epsilon_{t}\right\}$ is a white noise sequence with $E \epsilon_{t}=0$ and $E \epsilon_{t}^{2}=v$. Given $\left(x_{0}, y_{0}\right)$, we wish to choose $\left\{u_{t}\right\}_{t=0}^{9}$ to minimize $C=E\left[x_{10}^{2}+\sum_{t=0}^{9} u_{t}^{2}\right]$.

  Show that for some $\left\{a_{t}\right\}$ this problem can be reduced to one of controlling a scalar state $\xi_{t}=x_{t}+a_{t} y_{t}$.

  Find, in terms of $x_{0}, y_{0}$, the optimal $u_{0}$. What is the change in minimum $C$ achievable when the system starts in $\left(x_{0}, y_{0}\right)$ as compared to when it starts in $(0,0)$ ?

  Consider now a trajectory starting at $\left(x_{-1}, y_{-1}\right)=(11,-1)$. What value of $u_{-1}$ is optimal if we wish to minimize $5 u_{-1}^{2}+C$ ?

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• Paper 2, Section II, K

  #Optimization and Control

  2013

  Suppose $\left\{x_{t}\right\}_{t \geqslant 0}$ is a Markov chain. Consider the dynamic programming equation

  $$F_{s}(x)=\max \left\{r(x), \beta E\left[F_{s-1}\left(x_{1}\right) \mid x_{0}=x\right]\right\}, \quad s=1,2, \ldots,$$

  with $r(x)>0, \beta \in(0,1)$, and $F_{0}(x)=0$. Prove that:

  (i) $F_{s}(x)$ is nondecreasing in $s$;

  (ii) $F_{s}(x) \leqslant F(x)$, where $F(x)$ is the value function of an infinite-horizon problem that you should describe;

  (iii) $F_{\infty}(x)=\lim _{s \rightarrow \infty} F_{s}(x)=F(x)$.

  A coin lands heads with probability $p$. A statistician wishes to choose between: $H_{0}: p=1 / 3$ and $H_{1}: p=2 / 3$, one of which is true. Prior probabilities of $H_{1}$ and $H_{0}$ in the ratio $x: 1$ change after one toss of the coin to ratio $2 x: 1$ (if the toss was a head) or to ratio $x: 2$ (if the toss was a tail). What problem is being addressed by the following dynamic programming equation?

  $$F(x)=\max \left\{\frac{1}{1+x}, \frac{x}{1+x}, \beta\left[\left(\frac{1}{1+x} \frac{2}{3}+\frac{x}{1+x} \frac{1}{3}\right) F(x / 2)+\left(\frac{1}{1+x} \frac{1}{3}+\frac{x}{1+x} \frac{2}{3}\right) F(2 x)\right]\right\}$$

  Prove that $G(x)=(1+x) F(x)$ is a convex function of $x$.

  By sketching a graph of $G$, describe the form of the optimal policy.

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• Paper 4, Section II, C

  #Partial Differential Equations

  2013

  (i) Show that an arbitrary $C^{2}$ solution of the one-dimensional wave equation $u_{t t}-u_{x x}=0$ can be written in the form $u=F(x-t)+G(x+t)$.

  Hence, deduce the formula for the solution at arbitrary $t>0$ of the Cauchy problem

  $$u_{t t}-u_{x x}=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x)$$

  where $u_{0}, u_{1}$ are arbitrary Schwartz functions.

  Deduce from this formula a theorem on finite propagation speed for the onedimensional wave equation.

  (ii) Define the Fourier transform of a tempered distribution. Compute the Fourier transform of the tempered distribution $T_{t} \in \mathcal{S}^{\prime}(\mathbb{R})$ defined for all $t>0$ by the function

  $$T_{t}(y)= \begin{cases}\frac{1}{2} & \text { if }|y| \leqslant t \\ 0 & \text { if }|y|>t\end{cases}$$

  that is, $\left\langle T_{t}, f\right\rangle=\frac{1}{2} \int_{-t}^{+t} f(y) d y$ for all $f \in \mathcal{S}(\mathbb{R})$. By considering the Fourier transform in $x$, deduce from this the formula for the solution of $(*)$ that you obtained in part (i) in the case $u_{0}=0$.

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• Paper 3, Section II, C

  #Partial Differential Equations

  2013

  Define the parabolic boundary $\partial_{p a r} \Omega_{T}$ of the domain $\Omega_{T}=[0,1] \times(0, T]$ for $T>0$.

  Let $u=u(x, t)$ be a smooth real-valued function on $\Omega_{T}$ which satisfies the inequality

  $$u_{t}-a u_{x x}+b u_{x}+c u \leqslant 0$$

  Assume that the coefficients $a, b$ and $c$ are smooth functions and that there exist positive constants $m, M$ such that $m \leqslant a \leqslant M$ everywhere, and $c \geqslant 0$. Prove that

  $$\max _{(x, t) \in \bar{\Omega}_{T}} u(x, t) \leqslant \max _{(x, t) \in \partial_{\text {par }} \Omega_{T}} u^{+}(x, t) .$$

  [Here $u^{+}=\max \{u, 0\}$ is the positive part of the function $u$.]

  Consider a smooth real-valued function $\phi$ on $\Omega_{T}$ such that

  $$\phi_{t}-\phi_{x x}-\left(1-\phi^{2}\right) \phi=0, \quad \phi(x, 0)=f(x)$$

  everywhere, and $\phi(0, t)=1=\phi(1, t)$ for all $t \geqslant 0$. Deduce from $(*)$ that if $f(x) \leqslant 1$ for all $x \in[0,1]$ then $\phi(x, t) \leqslant 1$ for all $(x, t) \in \Omega_{T}$. [Hint: Consider $u=\phi^{2}-1$ and compute $\left.u_{t}-u_{x x} \cdot\right]$

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• Paper 1, Section II, C

  #Partial Differential Equations

  2013

  (i) Discuss briefly the concept of well-posedness of a Cauchy problem for a partial differential equation.

  Solve the Cauchy problem

  $$\partial_{2} u+x_{1} \partial_{1} u=a u^{2}, \quad u\left(x_{1}, 0\right)=\phi\left(x_{1}\right),$$

  where $a \in \mathbb{R}, \phi \in C^{1}(\mathbb{R})$ and $\partial_{i}$ denotes the partial derivative with respect to $x_{i}$ for $i=1,2$.

  For the case $a=0$ show that the solution satisfies $\max _{x_{1} \in \mathbb{R}}\left|u\left(x_{1}, x_{2}\right)\right|=\|\phi\|_{C^{0}}$, where the $C^{r}$ norm on functions $\phi=\phi\left(x_{1}\right)$ of one variable is defined by

  $$\|\phi\|_{C^{r}}=\sum_{i=0}^{r} \max _{x \in \mathbb{R}}\left|\partial_{1}^{i} \phi\left(x_{1}\right)\right|$$

  Deduce that the Cauchy problem is then well-posed in the uniform metric (i.e. the metric determined by the $C^{0}$ norm).

  (ii) State the Cauchy-Kovalevskaya theorem and deduce that the following Cauchy problem for the Laplace equation,

  $$\partial_{1}^{2} u+\partial_{2}^{2} u=0, \quad u\left(x_{1}, 0\right)=0, \partial_{2} u\left(x_{1}, 0\right)=\phi\left(x_{1}\right)$$

  has a unique analytic solution in some neighbourhood of $x_{2}=0$ for any analytic function $\phi=\phi\left(x_{1}\right)$. Write down the solution for the case $\phi\left(x_{1}\right)=\sin \left(n x_{1}\right)$, and hence give a sequence of initial data $\left\{\phi_{n}\left(x_{1}\right)\right\}_{n=1}^{\infty}$ with the property that

  $$\left\|\phi_{n}\right\|_{C^{r}} \rightarrow 0, \quad \text { as } n \rightarrow \infty, \text { for each } r \in \mathbb{N},$$

  whereas $u_{n}$, the corresponding solution of $(*)$, satisfies

  $$\max _{x_{1} \in \mathbb{R}}\left|u_{n}\left(x_{1}, x_{2}\right)\right| \rightarrow+\infty, \quad \text { as } n \rightarrow \infty$$

  for any $x_{2} \neq 0$.

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• Paper 2, Section II, C

  #Partial Differential Equations

  2013

  State the Lax-Milgram lemma.

  Let $\mathbf{V}=\mathbf{V}\left(x_{1}, x_{2}, x_{3}\right)$ be a smooth vector field which is $2 \pi$-periodic in each coordinate $x_{j}$ for $j=1,2,3$. Write down the definition of a weak $H_{p e r}^{1}$ solution for the equation

  $$-\Delta u+\sum_{j} V_{j} \partial_{j} u+u=f$$

  to be solved for $u=u\left(x_{1}, x_{2}, x_{3}\right)$ given $f=f\left(x_{1}, x_{2}, x_{3}\right)$ in $H^{0}$, with both $u$ and $f$ also $2 \pi$-periodic in each co-ordinate. [In this question use the definition

  $$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}^{3}} \hat{u}(m) e^{i m \cdot x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}^{3}}\left(1+\|m\|^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  for the Sobolev spaces of functions $2 \pi$-periodic in each coordinate $x_{j}$ and for $\left.s=0,1,2, \ldots\right]$

  If the vector field is divergence-free, prove that there exists a unique weak $H_{p e r}^{1}$ solution for all such $f$.

  Supposing that $\mathbf{V}$ is the constant vector field with components $(1,0,0)$, write down the solution of $(*)$ in terms of Fourier series and show that there exists $C>0$ such that

  $$\|u\|_{H^{2}} \leqslant C\|f\|_{H^{0}}$$

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• Paper 4, Section II, E

  #Principles of Quantum Mechanics

  2013

  (i) The creation and annihilation operators for a harmonic oscillator of angular frequency $\omega$ satisfy the commutation relation $\left[a, a^{\dagger}\right]=1$. Write down an expression for the Hamiltonian $H$ and number operator $N$ in terms of $a$ and $a^{\dagger}$. Explain how the space of eigenstates $|n\rangle, n=0,1,2, \ldots$, of $H$ is formed, and deduce the eigenenergies for these states. Show that

  $$a|n\rangle=\sqrt{n}|n-1\rangle, \quad a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle$$

  (ii) The operator $K_{r}$ is defined to be

  $$K_{r}=\frac{\left(a^{\dagger}\right)^{r} a^{r}}{r !}$$

  for $r=0,1,2, \ldots$ Show that $K_{r}$ commutes with $N$. Show that if $r \leqslant n$, then

  $$K_{r}|n\rangle=\frac{n !}{(n-r) ! r !}|n\rangle,$$

  and $K_{r}|n\rangle=0$ otherwise. By considering the action of $K_{r}$ on the state $|n\rangle$, deduce that

  $$\sum_{r=0}^{\infty}(-1)^{r} K_{r}=|0\rangle\langle 0|$$

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• Paper 3, Section II, E

  #Principles of Quantum Mechanics

  2013

  A particle moves in one dimension in an infinite square-well potential $V(x)=0$ for $|x|<a$ and $\infty$ for $|x|>a$. Find the energy eigenstates. Show that the energy eigenvalues are given by $E_{n}=E_{1} n^{2}$ for integer $n$, where $E_{1}$ is a constant which you should find.

  The system is perturbed by the potential $\delta V=\epsilon x / a$. Show that the energy of the $n^{\text {th }}$ level $E_{n}$ remains unchanged to first order in $\epsilon$. Show that the ground-state wavefunction is

  $$\psi_{1}(x)=\frac{1}{\sqrt{a}}\left[\cos \frac{\pi x}{2 a}+\frac{D \epsilon}{\pi^{2} E_{1}} \sum_{n=2,4, \ldots}(-1)^{A n} \frac{n^{B}}{\left(n^{2}-1\right)^{C}} \sin \frac{n \pi x}{2 a}+\mathcal{O}\left(\epsilon^{2}\right)\right]$$

  where $A, B, C$ and $D$ are numerical constants which you should find. Briefly comment on the conservation of parity in the unperturbed and perturbed systems.

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• Paper 2, Section II, 33E

  #Principles of Quantum Mechanics

  2013

  (i) In units where $\hbar=1$, angular momentum states $|j m\rangle$ obey

  $$J^{2}|j m\rangle=j(j+1)|j m\rangle, \quad J_{3}|j m\rangle=m|j m\rangle$$

  Use the algebra of angular momentum $\left[J_{i}, J_{j}\right]=i \epsilon_{i j k} J_{k}$ to derive the following in terms of $J^{2}, J_{\pm}=J_{1} \pm i J_{2}$ and $J_{3}$ : (a) $\left[J^{2}, J_{i}\right]$; (b) $\left[J_{3}, J_{\pm}\right]$; (c) $\left[J^{2}, J_{\pm}\right]$.

  (ii) Find $J_{+} J_{-}$in terms of $J^{2}$ and $J_{3}$. Thus calculate the quantum numbers of the state $J_{\pm}|j m\rangle$ in terms of $j$ and $m$. Derive the normalisation of the state $J_{-}|j m\rangle$. Therefore, show that

  $$\left\langle j j-1\left|J_{+}^{j-1} J_{-}^{j}\right| j j\right\rangle=\sqrt{A}(2 j-1) !$$

  finding $A$ in terms of $j$.

  (iii) Consider the combination of a spinless particle with an electron of spin $1 / 2$ and orbital angular momentum 1. Calculate the probability that the electron has a spin of $+1 / 2$ in the $z$-direction if the combined system has an angular momentum of $+1 / 2$ in the $z$-direction and a total angular momentum of $+3 / 2$. Repeat the calculation for a total angular momentum of $+1 / 2$.

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• Paper 1, Section II, E

  #Principles of Quantum Mechanics

  2013

  Consider a composite system of several identical particles. Describe how the multiparticle state is constructed from single-particle states. For the case of two identical particles, describe how considering the interchange symmetry leads to the definition of bosons and fermions.

  Consider two non-interacting, identical particles, each with spin 1 . The singleparticle, spin-independent Hamiltonian $H\left(\hat{\mathbf{x}}_{i}, \hat{\mathbf{p}}_{i}\right)$ has non-degenerate eigenvalues $E_{n}$ and wavefunctions $\psi_{n}\left(\mathbf{x}_{i}\right)$ where $i=1,2$ labels the particle and $n=0,1,2,3, \ldots$ In terms of these single-particle wavefunctions and single-particle spin states $|1\rangle,|0\rangle$ and $|-1\rangle$, write down all of the two-particle states and energies for:

  (i) the ground state;

  (ii) the first excited state.

  Assume now that $E_{n}$ is a linear function of $n$. Find the degeneracy of the $N^{\text {th }}$ energy level of the two-particle system for:

  (iii) $N$ even;

  (iv) $N$ odd.

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• Paper 4, Section II, $27 \mathrm{~K}$

  #Principles of Statistics

  2013

  Assuming only the existence and properties of the univariate normal distribution, define $\mathcal{N}_{p}(\underline{\mu}, \Sigma)$, the multivariate normal distribution with mean (row-)vector $\underline{\mu}$ and dispersion matrix $\Sigma$; and $W_{p}(\nu ; \Sigma)$, the Wishart distribution on integer $\nu>1$ degrees of freedom and with scale parameter $\Sigma$. Show that, if $\underline{X} \sim \mathcal{N}_{p}(\underline{\mu}, \Sigma), S \sim W_{p}(\nu ; \Sigma)$, and $\underline{b}(1 \times q), A(p \times q)$ are fixed, then $\underline{b}+\underline{X} A \sim \mathcal{N}_{q}(\underline{b}+\underline{\mu} A, \Phi), A^{\mathrm{T}} S A \sim W_{p}(\nu ; \Phi)$, where $\Phi=A^{\mathrm{T}} \Sigma A$.

  The random $(n \times p)$ matrix $X$ has rows that are independently distributed as $\mathcal{N}_{p}(\underline{\mathrm{M}}, \Sigma)$, where both parameters $\underline{\mathrm{M}}$ and $\Sigma$ are unknown. Let $\underline{\bar{X}}:=n^{-1} \mathbf{1}^{\mathrm{T}} X$, where 1 is the $(n \times 1)$ vector of $1 \mathrm{~s}$; and $S^{c}:=X^{\mathrm{T}} \Pi X$, with $\Pi:=I_{n}-n^{-1} 11^{\mathrm{T}}$. State the joint distribution of $\bar{X}$ and $S^{c}$ given the parameters.

  Now suppose $n>p$ and $\Sigma$ is positive definite. Hotelling's $T^{2}$ is defined as

  $$T^{2}:=n(\underline{\bar{X}}-\underline{\mathrm{M}})\left(\bar{S}^{c}\right)^{-1}(\underline{\bar{X}}-\underline{\mathrm{M}})^{\mathrm{T}}$$

  where $\bar{S}^{c}:=S^{c} / \nu$ with $\nu:=(n-1)$. Show that, for any values of $\underline{\mathrm{M}}$ and $\Sigma$,

  $$\left(\frac{\nu-p+1}{\nu p}\right) T^{2} \sim F_{\nu-p+1}^{p},$$

  the $F$ distribution on $p$ and $\nu-p+1$ degrees of freedom.

  [You may assume that:

  1. If $S \sim W_{p}(\nu ; \Sigma)$ and $\mathbf{a}$ is a fixed $(p \times 1)$ vector, then

  $$\frac{\mathbf{a}^{\mathrm{T}} \Sigma^{-1} \mathbf{a}}{\mathbf{a}^{\mathrm{T}} S^{-1} \mathbf{a}} \sim \chi_{\nu-p+1}^{2}$$

  2. If $V \sim \chi_{p}^{2}, W \sim \chi_{\lambda}^{2}$ are independent, then

  $$\frac{V / p}{W / \lambda} \sim F_{\lambda}^{p} \text {. }$$

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• Paper 3, Section II, K

  #Principles of Statistics

  2013

  What is meant by a convex decision problem? State and prove a theorem to the effect that, in a convex decision problem, there is no point in randomising. [You may use standard terms without defining them.]

  The sample space, parameter space and action space are each the two-point set $\{1,2\}$. The observable $X$ takes value 1 with probability $2 / 3$ when the parameter $\Theta=1$, and with probability $3 / 4$ when $\Theta=2$. The loss function $L(\theta, a)$ is 0 if $a=\theta$, otherwise 1 . Describe all the non-randomised decision rules, compute their risk functions, and plot these as points in the unit square. Identify an inadmissible non-randomised decision rule, and a decision rule that dominates it.

  Show that the minimax rule has risk function $(8 / 17,8 / 17)$, and is Bayes against a prior distribution that you should specify. What is its Bayes risk? Would a Bayesian with this prior distribution be bound to use the minimax rule?

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• Paper 1, Section II, K

  #Principles of Statistics

  2013

  When the real parameter $\Theta$ takes value $\theta$, variables $X_{1}, X_{2}, \ldots$ arise independently from a distribution $P_{\theta}$ having density function $p_{\theta}(x)$ with respect to an underlying measure $\mu$. Define the score variable $U_{n}(\theta)$ and the information function $I_{n}(\theta)$ for estimation of $\Theta$ based on $\boldsymbol{X}^{n}:=\left(X_{1}, \ldots, X_{n}\right)$, and relate $I_{n}(\theta)$ to $i(\theta):=I_{1}(\theta)$.

  State and prove the Cramér-Rao inequality for the variance of an unbiased estimator of $\Theta$. Under what conditions does this inequality become an equality? What is the form of the estimator in this case? [You may assume $\mathbb{E}_{\theta}\left\{U_{n}(\theta)\right\}=0, \operatorname{var}_{\theta}\left\{U_{n}(\theta)\right\}=I_{n}(\theta)$, and any further required regularity conditions, without comment.]

  Let $\widehat{\Theta}_{n}$ be the maximum likelihood estimator of $\Theta$ based on $\boldsymbol{X}^{n}$. What is the asymptotic distribution of $n^{\frac{1}{2}}\left(\widehat{\Theta}_{n}-\Theta\right)$ when $\Theta=\theta$ ?

  Suppose that, for each $n, \widehat{\Theta}_{n}$ is unbiased for $\Theta$, and the variance of $n^{\frac{1}{2}}\left(\widehat{\Theta}_{n}-\Theta\right)$ is exactly equal to its asymptotic variance. By considering the estimator $\alpha \widehat{\Theta}_{k}+(1-\alpha) \widehat{\Theta}_{n}$, or otherwise, show that, for $k<n, \operatorname{cov}_{\theta}\left(\widehat{\Theta}_{k}, \widehat{\Theta}_{n}\right)=\operatorname{var}_{\theta}\left(\widehat{\Theta}_{n}\right)$.

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• Paper 2, Section II, K

  #Principles of Statistics

  2013

  Describe the Weak Sufficiency Principle (WSP) and the Strong Sufficiency Principle (SSP). Show that Bayesian inference with a fixed prior distribution respects WSP.

  A parameter $\Phi$ has a prior distribution which is normal with mean 0 and precision (inverse variance) $h_{\Phi} .$ Given $\Phi=\phi$, further parameters $\Theta:=\left(\Theta{ }_{i}: i=1, \ldots, I\right)$ have independent normal distributions with mean $\phi$ and precision $h_{\Theta}$. Finally, given both $\Phi=\phi$ and $\boldsymbol{\Theta}=\boldsymbol{\theta}:=\left(\theta_{1}, \ldots, \theta_{I}\right)$, observables $\boldsymbol{X}:=\left(X_{i j}: i=1, \ldots, I ; j=1, \ldots, J\right)$ are independent, $X_{i j}$ being normal with mean $\theta_{i}$, and precision $h_{X}$. The precision parameters $\left(h_{\Phi}, h_{\Theta}, h_{X}\right)$ are all fixed and known. Let $\overline{\boldsymbol{X}}:=\left(\bar{X}_{1}, \ldots, \bar{X}_{I}\right)$, where $\bar{X}_{i}:=\sum_{j=1}^{J} X_{i j} / J$. Show, directly from the definition of sufficiency, that $\overline{\boldsymbol{X}}$ is sufficient for $(\Phi, \Theta)$. [You may assume without proof that, if $Y_{1}, \ldots, Y_{n}$ have independent normal distributions with the same variance, and $\bar{Y}:=n^{-1} \sum_{i=1}^{n} Y_{i}$, then the vector $\left(Y_{1}-\bar{Y}, \ldots, Y_{n}-\bar{Y}\right)$ is independent of $\bar{Y}$.]

  For data-values $\boldsymbol{x}:=\left(x_{i j}: i=1, \ldots, I ; j=1, \ldots, J\right)$, determine the joint distribution, $\Pi_{\phi}$ say, of $\Theta$, given $\boldsymbol{X}=\boldsymbol{x}$ and $\Phi=\phi .$ What is the distribution of $\Phi$, given $\boldsymbol{\Theta}=\boldsymbol{\theta}$ and $\boldsymbol{X}=\boldsymbol{x} ?$

  Using these results, describe clearly how Gibbs sampling combined with RaoBlackwellisation could be applied to estimate the posterior joint distribution of $\Theta$, given $\boldsymbol{X}=\boldsymbol{x}$.

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• Paper 4, Section II, K

  #Probability and Measure

  2013

  State Birkhoff's almost-everywhere ergodic theorem.

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables such that

  $$\mathbb{P}\left(X_{n}=0\right)=\mathbb{P}\left(X_{n}=1\right)=1 / 2$$

  Define for $k \in \mathbb{N}$

  $$Y_{k}=\sum_{n=1}^{\infty} X_{k+n-1} / 2^{n}$$

  What is the distribution of $Y_{k} ? \quad$ Show that the random variables $Y_{1}$ and $Y_{2}$ are not independent.

  Set $S_{n}=Y_{1}+\cdots+Y_{n}$. Show that $S_{n} / n$ converges as $n \rightarrow \infty$ almost surely and determine the limit. [You may use without proof any standard theorem provided you state it clearly.]

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• Paper 3, Section II, $25 K$

  #Probability and Measure

  2013

  Let $X$ be an integrable random variable with $\mathbb{E}(X)=0$. Show that the characteristic function $\phi_{X}$ is differentiable with $\phi_{X}^{\prime}(0)=0$. [You may use without proof standard convergence results for integrals provided you state them clearly.]

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all having the same distribution as $X$. Set $S_{n}=X_{1}+\cdots+X_{n}$. Show that $S_{n} / n \rightarrow 0$ in distribution. Deduce that $S_{n} / n \rightarrow 0$ in probability. [You may not use the Strong Law of Large Numbers.]

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• Paper 2, Section II, $26 \mathrm{~K}$

  #Probability and Measure

  2013

  Let $\left(f_{n}: n \in \mathbb{N}\right)$ be a sequence of non-negative measurable functions defined on a measure space $(E, \mathcal{E}, \mu)$. Show that $\liminf _{n} f_{n}$ is also a non-negative measurable function.

  State the Monotone Convergence Theorem.

  State and prove Fatou's Lemma.

  Let $\left(f_{n}: n \in \mathbb{N}\right)$ be as above. Suppose that $f_{n}(x) \rightarrow f(x)$ as $n \rightarrow \infty$ for all $x \in E$. Show that

  $$\mu\left(\min \left\{f_{n}, f\right\}\right) \rightarrow \mu(f) .$$

  Deduce that, if $f$ is integrable and $\mu\left(f_{n}\right) \rightarrow \mu(f)$, then $f_{n}$ converges to $f$ in $L^{1}$. [Still assume that $f_{n}$ and $f$ are as above.]

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• Paper 1, Section II, $26 \mathrm{~K}$

  #Probability and Measure

  2013

  State Dynkin's $\pi$-system $/ d$-system lemma.

  Let $\mu$ and $\nu$ be probability measures on a measurable space $(E, \mathcal{E})$. Let $\mathcal{A}$ be a $\pi$-system on $E$ generating $\mathcal{E}$. Suppose that $\mu(A)=\nu(A)$ for all $A \in \mathcal{A}$. Show that $\mu=\nu$.

  What does it mean to say that a sequence of random variables is independent?

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all uniformly distributed on $[0,1]$. Let $Y$ be another random variable, independent of $\left(X_{n}: n \in \mathbb{N}\right)$. Define random variables $Z_{n}$ in $[0,1]$ by $Z_{n}=\left(X_{n}+Y\right) \bmod 1$. What is the distribution of $Z_{1}$ ? Justify your answer.

  Show that the sequence of random variables $\left(Z_{n}: n \in \mathbb{N}\right)$ is independent.

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• Paper 3, Section II, G

  #Representation Theory

  2013

  Suppose that $\left(\rho_{1}, V_{1}\right)$ and $\left(\rho_{2}, V_{2}\right)$ are complex representations of the finite groups $G_{1}$ and $G_{2}$ respectively. Use $\rho_{1}$ and $\rho_{2}$ to construct a representation $\rho_{1} \otimes \rho_{2}$ of $G_{1} \times G_{2}$ on $V_{1} \otimes V_{2}$ and show that its character satisfies

  $$\chi \rho_{1} \otimes \rho_{2}\left(g_{1}, g_{2}\right)=\chi_{\rho_{1}}\left(g_{1}\right) \chi \rho_{2}\left(g_{2}\right)$$

  for each $g_{1} \in G_{1}, g_{2} \in G_{2}$.

  Prove that if $\rho_{1}$ and $\rho_{2}$ are irreducible then $\rho_{1} \otimes \rho_{2}$ is irreducible as a representation of $G_{1} \times G_{2}$. Moreover, show that every irreducible complex representation of $G_{1} \times G_{2}$ arises in this way.

  Is it true that every complex representation of $G_{1} \times G_{2}$ is of the form $\rho_{1} \otimes \rho_{2}$ with $\rho_{i}$ a complex representation of $G_{i}$ for $i=1,2 ?$ Justify your answer.

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• Paper 2, Section II, G

  #Representation Theory

  2013

  Recall that a regular icosahedron has 20 faces, 30 edges and 12 vertices. Let $G$ be the group of rotational symmetries of a regular icosahedron.

  Compute the conjugacy classes of $G$. Hence, or otherwise, construct the character table of $G$. Using the character table explain why $G$ must be a simple group.

  [You may use any general theorems provided that you state them clearly.]

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• Paper 4, Section II, G

  #Representation Theory

  2013

  State and prove Burnside's $p^{a} q^{b}$-theorem.

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• Paper 1, Section II, 19G

  #Representation Theory

  2013

  State and prove Maschke's Theorem for complex representations of finite groups.

  Without using character theory, show that every irreducible complex representation of the dihedral group of order $10, D_{10}$, has dimension at most two. List the irreducible complex representations of $D_{10}$ up to isomorphism.

  Let $V$ be the set of vertices of a regular pentagon with the usual action of $D_{10}$. Explicitly decompose the permutation representation $\mathbb{C} V$ into a direct sum of irreducible subrepresentations.

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• Paper 3, Section II, I

  #Riemann Surfaces

  2013

  Let $\Lambda=\mathbb{Z}+\mathbb{Z} \lambda$ be a lattice in $\mathbb{C}$ where $\operatorname{Im}(\lambda)>0$, and let $X$ be the complex torus $\mathbb{C} / \Lambda .$

  (i) Give the definition of an elliptic function with respect to $\Lambda$. Show that there is a bijection between the set of elliptic functions with respect to $\Lambda$ and the set of holomorphic maps from $X$ to the Riemann sphere. Next, show that if $f$ is an elliptic function with respect to $\Lambda$ and $f^{-1}\{\infty\}=\emptyset$, then $f$ is constant.

  (ii) Assume that

  $$f(z)=\frac{1}{z^{2}}+\sum_{\omega \in \Lambda \backslash\{0\}}\left(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right)$$

  defines a meromorphic function on $\mathbb{C}$, where the sum converges uniformly on compact subsets of $\mathbb{C} \backslash \Lambda$. Show that $f$ is an elliptic function with respect to $\Lambda$. Calculate the order of $f$.

  Let $g$ be an elliptic function with respect to $\Lambda$ on $\mathbb{C}$, which is holomorphic on $\mathbb{C} \backslash \Lambda$ and whose only zeroes in the closed parallelogram with vertices $\{0,1, \lambda, \lambda+1\}$ are simple zeroes at the points $\left\{\frac{1}{2}, \frac{\lambda}{2}, \frac{1}{2}+\frac{\lambda}{2}\right\}$. Show that $g$ is a non-zero constant multiple of $f^{\prime}$.

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• Paper 2, Section II, I

  #Riemann Surfaces

  2013

  (i) Show that the open unit $\operatorname{disc} D=\{z \in \mathbb{C}:|z|<1\}$ is biholomorphic to the upper half-plane $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$.

  (ii) Define the degree of a non-constant holomorphic map between compact connected Riemann surfaces. State the Riemann-Hurwitz formula without proof. Now let $X$ be a complex torus and $f: X \rightarrow Y$ a holomorphic map of degree 2 , where $Y$ is the Riemann sphere. Show that $f$ has exactly four branch points.

  (iii) List without proof those Riemann surfaces whose universal cover is the Riemann sphere or $\mathbb{C}$. Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic map such that there are two distinct elements $a, b \in \mathbb{C}$ outside the image of $f$. Assuming the uniformization theorem and the monodromy theorem, show that $f$ is constant.

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• Paper 1, Section II, I

  #Riemann Surfaces

  2013

  (i) Let $f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with radius of convergence $r$ in $(0, \infty)$. Show that there is at least one point $a$ on the circle $C=\{z \in \mathbb{C}:|z|=r\}$ which is a singular point of $f$, that is, there is no direct analytic continuation of $f$ in any neighbourhood of $a$.

  (ii) Let $X$ and $Y$ be connected Riemann surfaces. Define the space $\mathcal{G}$ of germs of function elements of $X$ into $Y$. Define the natural topology on $\mathcal{G}$ and the natural $\operatorname{map} \pi: \mathcal{G} \rightarrow X$. [You may assume without proof that the topology on $\mathcal{G}$ is Hausdorff.] Show that $\pi$ is continuous. Define the natural complex structure on $\mathcal{G}$ which makes it into a Riemann surface. Finally, show that there is a bijection between the connected components of $\mathcal{G}$ and the complete holomorphic functions of $X$ into $Y$.

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• Paper 4, Section I, J

  #Statistical Modelling

  2013

  The output $X$ of a process depends on the levels of two adjustable variables: $A$, a factor with four levels, and $B$, a factor with two levels. For each combination of a level of $A$ and a level of $B$, nine independent values of $X$ are observed.

  Explain and interpret the $R$ commands and (abbreviated) output below. In particular, describe the model being fitted, and describe and comment on the hypothesis tests performed under the summary and anova commands.

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• Paper 3, Section I, J

  #Statistical Modelling

  2013

  Consider the linear model $Y=X \beta+\epsilon$ where $Y=\left(Y_{1}, \ldots, Y_{n}\right)^{\mathrm{T}}, \beta=\left(\beta_{1}, \ldots, \beta_{p}\right)^{\mathrm{T}}$, and $\epsilon=\left(\epsilon_{1}, \ldots, \epsilon_{n}\right)^{\mathrm{T}}$, with $\epsilon_{1}, \ldots, \epsilon_{n}$ independent $N\left(0, \sigma^{2}\right)$ random variables. The $(n \times p)$ matrix $X$ is known and is of full rank $p<n$. Give expressions for the maximum likelihood estimators $\widehat{\beta}$ and $\widehat{\sigma}^{2}$ of $\beta$ and $\sigma^{2}$ respectively, and state their joint distribution. Show that $\widehat{\beta}$ is unbiased whereas $\widehat{\sigma}^{2}$ is biased.

  Suppose that a new variable $Y^{*}$ is to be observed, satisfying the relationship

  $$Y^{*}=x^{* \mathrm{~T}} \beta+\epsilon^{*}$$

  where $x^{*}(p \times 1)$ is known, and $\epsilon^{*} \sim N\left(0, \sigma^{2}\right)$ independently of $\epsilon$. We propose to predict $Y^{*}$ by $\widetilde{Y}=x^{* \mathrm{~T}} \widehat{\beta}$. Identify the distribution of

  $$\frac{Y^{*}-\tilde{Y}}{\tau \tilde{\sigma}}$$

  where

  $$\begin{aligned} \tilde{\sigma}^{2} &=\frac{n}{n-p} \widehat{\sigma}^{2} \\ \tau^{2} &=x^{* \mathrm{~T}}\left(X^{\mathrm{T}} X\right)^{-1} x^{*}+1 \end{aligned}$$

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• Paper 2, Section I, J

  #Statistical Modelling

  2013

  Consider a linear model $Y=X \beta+\epsilon$, where $Y$ and $\epsilon$ are $(n \times 1)$ with $\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$, $\beta$ is $(p \times 1)$, and $X$ is $(n \times p)$ of full $\operatorname{rank} p<n$. Let $\gamma$ and $\delta$ be sub-vectors of $\beta$. What is meant by orthogonality between $\gamma$ and $\delta$ ?

  Now suppose

  $$Y_{i}=\beta_{0}+\beta_{1} x_{i}+\beta_{2} x_{i}^{2}+\beta_{3} P_{3}\left(x_{i}\right)+\epsilon_{i} \quad(i=1, \ldots, n),$$

  where $\epsilon_{1}, \ldots, \epsilon_{n}$ are independent $N\left(0, \sigma^{2}\right)$ random variables, $x_{1}, \ldots, x_{n}$ are real-valued known explanatory variables, and $P_{3}(x)$ is a cubic polynomial chosen so that $\beta_{3}$ is orthogonal to $\left(\beta_{0}, \beta_{1}, \beta_{2}\right)^{\mathrm{T}}$ and $\beta_{1}$ is orthogonal to $\left(\beta_{0}, \beta_{2}\right)^{\mathrm{T}}$.

  Let $\widetilde{\beta}=\left(\beta_{0}, \beta_{2}, \beta_{1}, \beta_{3}\right)^{\mathrm{T}}$. Describe the matrix $\widetilde{X}$ such that $Y=\widetilde{X} \widetilde{\beta}+\epsilon$. Show that $\widetilde{X}^{\mathrm{T}} \widetilde{X}$ is block diagonal. Assuming further that this matrix is non-singular, show that the least-squares estimators of $\beta_{1}$ and $\beta_{3}$ are, respectively,

  $$\widehat{\beta}_{1}=\frac{\sum_{i=1}^{n} x_{i} Y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} \quad \text { and } \quad \widehat{\beta}_{3}=\frac{\sum_{i=1}^{n} P_{3}\left(x_{i}\right) Y_{i}}{\sum_{i=1}^{n} P_{3}\left(x_{i}\right)^{2}}$$

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• Paper 1, Section I, J

  #Statistical Modelling

  2013

  Variables $Y_{1}, \ldots, Y_{n}$ are independent, with $Y_{i}$ having a density $p\left(y \mid \mu_{i}\right)$ governed by an unknown parameter $\mu_{i}$. Define the deviance for a model $M$ that imposes relationships between the $\left(\mu_{i}\right)$.

  From this point on, suppose $Y_{i} \sim \operatorname{Poisson}\left(\mu_{i}\right)$. Write down the log-likelihood of data $y_{1}, \ldots, y_{n}$ as a function of $\mu_{1}, \ldots, \mu_{n}$.

  Let $\widehat{\mu}_{i}$ be the maximum likelihood estimate of $\mu_{i}$ under model $M$. Show that the deviance for this model is given by

  $$2 \sum_{i=1}^{n}\left\{y_{i} \log \frac{y_{i}}{\widehat{\mu}_{i}}-\left(y_{i}-\widehat{\mu}_{i}\right)\right\}$$

  Now suppose that, under $M, \log \mu_{i}=\beta^{\mathrm{T}} x_{i}, i=1, \ldots, n$, where $x_{1}, \ldots, x_{n}$ are known $p$-dimensional explanatory variables and $\beta$ is an unknown $p$-dimensional parameter. Show that $\widehat{\mu}:=\left(\widehat{\mu}_{1}, \ldots, \widehat{\mu}_{n}\right)^{\mathrm{T}}$ satisfies $X^{\mathrm{T}} y=X^{\mathrm{T}} \widehat{\mu}$, where $y=\left(y_{1}, \ldots, y_{n}\right)^{\mathrm{T}}$ and $X$ is the $(n \times p)$ matrix with rows $x_{1}^{\mathrm{T}}, \ldots, x_{n}^{\mathrm{T}}$, and express this as an equation for the maximum likelihood estimate $\widehat{\beta}$ of $\beta$. [You are not required to solve this equation.]

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• Paper 4, Section II, J

  #Statistical Modelling

  2013

  Let $f_{0}$ be a probability density function, with cumulant generating function $K$. Define what it means for a random variable $Y$ to have a model function of exponential dispersion family form, generated by $f_{0}$.

  A random variable $Y$ is said to have an inverse Gaussian distribution, with parameters $\phi$ and $\lambda$ (both positive), if its density function is

  $$f(y ; \phi, \lambda)=\frac{\sqrt{\lambda}}{\sqrt{2 \pi y^{3}}} e^{\sqrt{\lambda \phi}} \exp \left\{-\frac{1}{2}\left(\frac{\lambda}{y}+\phi y\right)\right\} \quad(y>0)$$

  Show that the family of all inverse Gaussian distributions for $Y$ is of exponential dispersion family form. Deduce directly the corresponding expressions for $E(Y)$ and $\operatorname{Var}(Y)$ in terms of $\phi$ and $\lambda$. What are the corresponding canonical link function and variance function?

  Consider a generalized linear model, $M$, for independent variables $Y_{i}(i=1, \ldots, n)$, whose random component is defined by the inverse Gaussian distribution with link function $g(\mu)=\log (\mu):$ thus $g\left(\mu_{i}\right)=x_{i}^{\mathrm{T}} \beta$, where $\beta=\left(\beta_{1}, \ldots, \beta_{p}\right)^{\mathrm{T}}$ is the vector of unknown regression coefficients and $x_{i}=\left(x_{i 1}, \ldots, x_{i p}\right)^{\mathrm{T}}$ is the vector of known values of the explanatory variables for the $i^{\text {th }}$ observation. The vectors $x_{i}(i=1, \ldots, n)$ are linearly independent. Assuming that the dispersion parameter is known, obtain expressions for the score function and Fisher information matrix for $\beta$. Explain how these can be used to compute the maximum likelihood estimate $\widehat{\beta}$ of $\beta$.

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• Paper 1, Section II, J

  #Statistical Modelling

  2013

  A cricket ball manufacturing company conducts the following experiment. Every day, a bowling machine is set to one of three levels, "Medium", "Fast" or "Spin", and then bowls 100 balls towards the stumps. The number of times the ball hits the stumps and the average wind speed (in kilometres per hour) during the experiment are recorded, yielding the following data (abbreviated):

  $\begin{array}{llll}\text { Day } & \text { Wind } & \text { Level } & \text { Stumps } \\ 1 & 10 & \text { Medium } & 26 \\ 2 & 8 & \text { Medium } & 37 \\ \vdots & \vdots & \vdots & \vdots \\ 50 & 12 & \text { Medium } & 32 \\ 51 & 7 & \text { Fast } & 31 \\ \vdots & \vdots & \vdots & \vdots \\ 120 & 3 & \text { Fast } & 28 \\ 121 & 5 & \text { Spin } & 35 \\ \vdots & \vdots & \vdots & \vdots \\ 150 & 6 & \text { Spin } & 31\end{array}$

  Write down a reasonable model for $Y_{1}, \ldots, Y_{150}$, where $Y_{i}$ is the number of times the ball hits the stumps on the $i^{t h}$ day. Explain briefly why we might want to include interactions between the variables. Write $R$ code to fit your model.

  The company's statistician fitted her own generalized linear model using $\mathrm{R}$, and obtained the following summary (abbreviated):

  $\begin{array}{lrrrrr}\text { >summary(ball) } & & & & \\ \text { Coefficients: } & & & & & \\ & \text { Estimate } & \text { Std. Error } & \text { z value } & \operatorname{Pr}(>|z|) & \\ \text { (Intercept) } & -0.37258 & 0.05388 & -6.916 & 4.66 \mathrm{e}-12 & * * * \\ \text { Wind } & 0.09055 & 0.01595 & 5.676 & 1.38 \mathrm{e}-08 & * * * \\ \text { LevelFast } & -0.10005 & 0.08044 & -1.244 & 0.213570 & \\ \text { LevelSpin } & 0.29881 & 0.08268 & 3.614 & 0.000301 & * * * \\ \text { Wind: LevelFast } & 0.03666 & 0.02364 & 1.551 & 0.120933 & \\ \text { Wind: LevelSpin } & -0.07697 & 0.02845 & -2.705 & 0.006825 & * *\end{array}$

  Why are LevelMedium and Wind: LevelMedium not listed?

  Suppose that, on another day, the bowling machine is set to "Spin", and the wind speed is 5 kilometres per hour. What linear function of the parameters should the statistician use in constructing a predictor of the number of times the ball hits the stumps that day?

  Based on the above output, how might you improve the model? How could you fit your new model in $R$ ?

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• Paper 4, Section II, A

  #Statistical Physics

  2013

  A classical particle of mass $m$ moving non-relativistically in two-dimensional space is enclosed inside a circle of radius $R$ and attached by a spring with constant $\kappa$ to the centre of the circle. The particle thus moves in a potential

  $$V(r)= \begin{cases}\frac{1}{2} \kappa r^{2} & \text { for } r<R \\ \infty & \text { for } r \geqslant R\end{cases}$$

  where $r^{2}=x^{2}+y^{2}$. Let the particle be coupled to a heat reservoir at temperature $T$.

  (i) Which of the ensembles of statistical physics should be used to model the system?

  (ii) Calculate the partition function for the particle.

  (iii) Calculate the average energy $\langle E\rangle$ and the average potential energy $\langle V\rangle$ of the particle.

  (iv) What is the average energy in:

  (a) the limit $\frac{1}{2} \kappa R^{2} \gg k_{\mathrm{B}} T$ (strong coupling)?

  (b) the limit $\frac{1}{2} \kappa R^{2} \ll k_{\mathrm{B}} T$ (weak coupling)?

  Compare the two results with the values expected from equipartition of energy.

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• Paper 3, Section II, 35A

  #Statistical Physics

  2013

  (i) Briefly describe the microcanonical ensemble.

  (ii) For quantum mechanical systems the energy levels are discrete. Explain why we can write the probability distribution in this case as

  $$p\left(\left\{n_{i}\right\}\right)= \begin{cases}\text { const }>0 & \text { for } E \leqslant E\left(\left\{n_{i}\right\}\right)<E+\Delta E \\ 0 & \text { otherwise }\end{cases}$$

  What assumption do we make for the energy interval $\Delta E$ ?

  Consider $N$ independent linear harmonic oscillators of equal frequency $\omega$. Their total energy is given by

  $$E\left(\left\{n_{i}\right\}\right)=\sum_{i=1}^{N} \hbar \omega\left(n_{i}+\frac{1}{2}\right)=M \hbar \omega+\frac{N}{2} \hbar \omega \quad \text { with } \quad M=\sum_{i=1}^{N} n_{i}$$

  Here $n_{i}=0,1,2, \ldots$ is the excitation number of oscillator $i$.

  (iii) Show that, for fixed $N$ and $M$, the number $g_{N}(M)$ of possibilities to distribute the $M$ excitations over $N$ oscillators (i.e. the number of different choices $\left\{n_{i}\right\}$ consistent with $M$ ) is given by

  $$g_{N}(M)=\frac{(M+N-1) !}{M !(N-1) !}$$

  [Hint: You may wish to consider the set of $N$ oscillators plus $M-1$ "additional" excitations and what it means to choose $M$ objects from this set.]

  (iv) Using the probability distribution of part (ii), calculate the probability distribution $p\left(E_{1}\right)$ for the "first" oscillator as a function of its energy $E_{1}=n_{1} \hbar \omega+\frac{1}{2} \hbar \omega$.

  (v) If $\Delta E=\hbar \omega \ll E$ then exactly one value of $M$ will correspond to a total energy inside the interval $(E, E+\Delta E)$. In this case, show that

  $$p\left(E_{1}\right) \approx \frac{g_{N-1}\left(M-n_{1}\right)}{g_{N}(M)} .$$

  Approximate this result in the limit $N \gg 1, M \gg n_{1}$.

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• Paper 2, Section II, 35A

  #Statistical Physics

  2013

  (i) The first law of thermodynamics is $d E=T d S-p d V+\mu d N$, where $\mu$ is the chemical potential. Briefly describe its meaning.

  (ii) What is equipartition of energy? Under which conditions is it valid? Write down the heat capacity $C_{V}$ at constant volume for a monatomic ideal gas.

  (iii) Starting from the first law of thermodynamics, and using the fact that for an ideal gas $(\partial E / \partial V)_{T}=0$, show that the entropy of an ideal gas containing $N$ particles can be written as

  $$S(T, V)=N\left(\int \frac{c_{V}(T)}{T} d T+k_{\mathrm{B}} \ln \frac{V}{N}+\mathrm{const}\right)$$

  where $T$ and $V$ are temperature and volume of the gas, $k_{\mathrm{B}}$ is the Boltzmann constant, and we define the heat capacity per particle as $c_{V}=C_{V} / N$.

  (iv) The Gibbs free energy $G$ is defined as $G=E+p V-T S$. Verify that it is a function of temperature $T$, pressure $p$ and particle number $N$. Explain why $G$ depends on the particle number $N$ through $G=\mu(T, p) N$.

  (v) Calculate the chemical potential $\mu$ for an ideal gas with heat capacity per particle $c_{V}(T)$. Calculate $\mu$ for the special case of a monatomic gas.

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• Paper 1, Section II, 35A

  #Statistical Physics

  2013

  (i) What is the occupation number of a state $i$ with energy $E_{i}$ according to the Fermi-Dirac statistics for a given chemical potential $\mu$ ?

  (ii) Assuming that the energy $E$ is spin independent, what is the number $g_{s}$ of electrons which can occupy an energy level?

  (iii) Consider a semi-infinite metal slab occupying $z \leqslant 0$ (and idealized to have infinite extent in the $x y$ plane) and a vacuum environment at $z>0$. An electron with momentum $\left(p_{x}, p_{y}, p_{z}\right)$ inside the slab will escape the metal in the $+z$ direction if it has a sufficiently large momentum $p_{z}$ to overcome a potential barrier $V_{0}$ relative to the Fermi energy $\epsilon_{\mathrm{F}}$, i.e. if

  $$\frac{p_{z}^{2}}{2 m} \geqslant \epsilon_{\mathrm{F}}+V_{0}$$

  where $m$ is the electron mass.

  At fixed temperature $T$, some fraction of electrons will satisfy this condition, which results in a current density $j_{z}$ in the $+z$ direction (an electron having escaped the metal once is considered lost, never to return). Each electron escaping provides a contribution $\delta j_{z}=-e v_{z}$ to this current density, where $v_{z}$ is the velocity and $e$ the elementary charge.

  (a) Briefly describe the Fermi-Dirac distribution as a function of energy in the limit $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$, where $k_{\mathrm{B}}$ is the Boltzmann constant. What is the chemical potential $\mu$ in this limit?

  (b) Assume that the electrons behave like an ideal, non-relativistic Fermi gas and that $k_{\mathrm{B}} T \ll V_{0}$ and $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$. Calculate the current density $j_{z}$ associated with the electrons escaping the metal in the $+z$ direction. How could we easily increase the strength of the current?

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• Paper 4, Section II, J

  #Stochastic Financial Models

  2013

  Let $S_{t}:=\left(S_{t}^{1}, S_{t}^{2}, \ldots, S_{t}^{n}\right)^{\mathrm{T}}$ denote the time- $t$ prices of $n$ risky assets in which an agent may invest, $t=0,1$. He may also invest his money in a bank account, which will return interest at rate $r>0$. At time 0 , he knows $S_{0}$ and $r$, and he knows that $S_{1} \sim N(\mu, V)$. If he chooses at time 0 to invest cash value $\theta_{i}$ in risky asset $i$, express his wealth $w_{1}$ at time 1 in terms of his initial wealth $w_{0}>0$, the choices $\theta:=\left(\theta_{1}, \ldots, \theta_{n}\right)^{\mathrm{T}}$, the value of $S_{1}$, and $r$.

  Suppose that his goal is to minimize the variance of $w_{1}$ subject to the requirement that the mean $E\left(w_{1}\right)$ should be at least $m$, where $m \geqslant(1+r) w_{0}$ is given. What portfolio $\theta$ should he choose to achieve this?

  Suppose instead that his goal is to minimize $E\left(w_{1}^{2}\right)$ subject to the same constraint. Show that his optimal portfolio is unchanged.

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• Paper 3, Section II, J

  #Stochastic Financial Models

  2013

  Suppose that $\left(\varepsilon_{t}\right)_{t=0,1, \ldots, T}$ is a sequence of independent and identically distributed random variables such that $E \exp \left(z \varepsilon_{1}\right)<\infty$ for all $z \in \mathbb{R}$. Each day, an agent receives an income, the income on day $t$ being $\varepsilon_{t}$. After receiving this income, his wealth is $w_{t}$. From this wealth, he chooses to consume $c_{t}$, and invests the remainder $w_{t}-c_{t}$ in a bank account which pays a daily interest rate of $r>0$. Write down the equation for the evolution of $w_{t}$.

  Suppose we are given constants $\beta \in(0,1), A_{T}, \gamma>0$, and define the functions

  $$U(x)=-\exp (-\gamma x), \quad U_{T}(x)=-A_{T} \exp (-\nu x)$$

  where $\nu:=\gamma r /(1+r)$. The agent's objective is to attain

  $$V_{0}(w):=\sup E\left\{\sum_{t=0}^{T-1} \beta^{t} U\left(c_{t}\right)+\beta^{T} U_{T}\left(w_{T}\right) \mid w_{0}=w\right\},$$

  where the supremum is taken over all adapted sequences $\left(c_{t}\right)$. If the value function is defined for $0 \leqslant n<T$ by

  $$V_{n}(w)=\sup E\left\{\sum_{t=n}^{T-1} \beta^{t-n} U\left(c_{t}\right)+\beta^{T-n} U_{T}\left(w_{T}\right) \mid w_{n}=w\right\}$$

  with $V_{T}=U_{T}$, explain briefly why you expect the $V_{n}$ to satisfy

  $$V_{n}(w)=\sup _{c}\left[U(c)+\beta E\left\{V_{n+1}\left((1+r)(w-c)+\varepsilon_{n+1}\right)\right\}\right]$$

  Show that the solution to $(*)$ has the form

  $$V_{n}(w)=-A_{n} \exp (-\nu w),$$

  for constants $A_{n}$ to be identified. What is the form of the consumption choices that achieve the supremum in $(*)$ ?

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• Paper 1, Section II, 29J

  #Stochastic Financial Models

  2013

  (i) Suppose that the price $S_{t}$ of an asset at time $t$ is given by

  $$S_{t}=S_{0} \exp \left\{\sigma B_{t}+\left(r-\frac{1}{2} \sigma^{2}\right) t\right\}$$

  where $B$ is a Brownian motion, $S_{0}$ and $\sigma$ are positive constants, and $r$ is the riskless rate of interest, assumed constant. In this model, explain briefly why the time-0 price of a derivative which delivers a bounded random variable $Y$ at time $T$ should be given by $E\left(e^{-r T} Y\right)$. What feature of this model ensures that the price is unique?

  Derive an expression $C\left(S_{0}, K, T, r, \sigma\right)$ for the time- 0 price of a European call option with strike $K$ and expiry $T$. Explain the italicized terms.

  (ii) Suppose now that the price $X_{t}$ of an asset at time $t$ is given by

  $$X_{t}=\sum_{j=1}^{n} w_{j} \exp \left\{\sigma_{j} B_{t}+\left(r-\frac{1}{2} \sigma_{j}^{2}\right) t\right\}$$

  where the $w_{j}$ and $\sigma_{j}$ are positive constants, and the other notation is as in part (i) above. Show that the time-0 price of a European call option with strike $K$ and expiry $T$ written on this asset can be expressed as

  $$\sum_{j=1}^{n} C\left(w_{j}, k_{j}, T, r, \sigma_{j}\right)$$

  where the $k_{j}$ are constants. Explain how the $k_{j}$ are characterized.

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• Paper 2, Section II, J

  #Stochastic Financial Models

  2013

  What does it mean to say that $\left(Y_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a supermartingale?

  State and prove Doob's Upcrossing Inequality for a supermartingale.

  Let $\left(M_{n}, \mathcal{F}_{n}\right)_{n \leqslant 0}$ be a martingale indexed by negative time, that is, for each $n \leqslant 0$, $\mathcal{F}_{n-1} \subseteq \mathcal{F}_{n}, M_{n} \in L^{1}\left(\mathcal{F}_{n}\right)$ and $E\left[M_{n} \mid \mathcal{F}_{n-1}\right]=M_{n-1}$. Using Doob's Upcrossing Inequality, prove that the limit $\lim _{n \rightarrow-\infty} M_{n}$ exists almost surely.

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• Paper 4, Section I, F

  #Topics in Analysis

  2013

  State the Baire Category Theorem. A set $X \subseteq \mathbb{R}$ is said to be a $G_{\delta}$-set if it is the intersection of countably many open sets. Show that the set $\mathbb{Q}$ of rationals is not a $G_{\delta}$-set.

  [You may assume that the rationals are countable and that $\mathbb{R}$ is complete.]

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• Paper 3, Section I, $2 F$

  #Topics in Analysis

  2013

  State Brouwer's fixed point theorem. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function with the property that $|f(x)-x| \leqslant 1$ for all $x$. Show that $f$ is surjective.

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• Paper 2, Section I, F

  #Topics in Analysis

  2013

  (i) Show that for every $\epsilon>0$ there is a polynomial $p: \mathbb{R} \rightarrow \mathbb{R}$ such that $\left|\frac{1}{x}-p(x)\right| \leqslant \epsilon$ for all $x \in \mathbb{R}$ satisfying $\frac{1}{2} \leqslant|x| \leqslant 2$.

  [You may assume standard results provided they are stated clearly.]

  (ii) Show that there is no polynomial $p: \mathbb{C} \rightarrow \mathbb{C}$ such that $\left|\frac{1}{z}-p(z)\right|<1$ for all $z \in \mathbb{C}$ satisfying $\frac{1}{2} \leqslant|z| \leqslant 2$.

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• Paper 1, Section I, $2 F$

  #Topics in Analysis

  2013

  Show that $\sin (1)$ is irrational. [The angle is measured in radians.]

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• Paper 2, Section II, F

  #Topics in Analysis

  2013

  (i) Let $n \geqslant 4$ be an integer. Show that

  $$1+\frac{1}{n+\frac{1}{1+\frac{1}{n+\ldots}}} \geqslant 1+\frac{1}{2 n} .$$

  (ii) Let us say that an irrational number $\alpha$ is badly approximable if there is some constant $c>0$ such that

  $$\left|\alpha-\frac{p}{q}\right| \geqslant \frac{c}{q^{2}}$$

  for all $q \geqslant 1$ and for all integers $p$. Show that if the integers $a_{n}$ in the continued fraction expansion $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ are bounded then $\alpha$ is badly approximable.

  Give, with proof, an example of an irrational number which is not badly approximable.

  [Standard facts about continued fractions may be used without proof provided they are stated clearly.]

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• Paper 3, Section II, F

  #Topics in Analysis

  2013

  Suppose that $x_{0}, x_{1}, \ldots, x_{n} \in[-1,1]$ are distinct points. Let $f$ be an infinitely differentiable real-valued function on an open interval containing $[-1,1]$. Let $p$ be the unique polynomial of degree at most $n$ such that $f\left(x_{r}\right)=p\left(x_{r}\right)$ for $r=0,1, \ldots, n$. Show that for each $x \in[-1,1]$ there is some $\xi \in(-1,1)$ such that

  $$f(x)-p(x)=\frac{f^{(n+1)}(\xi)}{(n+1) !}\left(x-x_{0}\right) \ldots\left(x-x_{n}\right)$$

  Now take $x_{r}=\cos \frac{2 r+1}{2 n+2} \pi$. Show that

  $$|f(x)-p(x)| \leqslant \frac{1}{2^{n}(n+1) !} \sup _{\xi \in[-1,1]}\left|f^{(n+1)}(\xi)\right|$$

  for all $x \in[-1,1]$. Deduce that there is a polynomial $p$ of degree at most $n$ such that

  $$\left|\frac{1}{3+x}-p(x)\right| \leqslant \frac{1}{4^{n+1}}$$

  for all $x \in[-1,1]$.

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• Paper 4, Section II, 38C

  #Waves

  2013

  A wave disturbance satisfies the equation

  $$\frac{\partial^{2} \psi}{\partial t^{2}}-c^{2} \frac{\partial^{2} \psi}{\partial x^{2}}+c^{2} \psi=0$$

  where $c$ is a positive constant. Find the dispersion relation, and write down the solution to the initial-value problem for which $\partial \psi / \partial t(x, 0)=0$ for all $x$, and $\psi(x, 0)$ is given in the form

  $$\psi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

  where $A(k)$ is a real function with $A(k)=A(-k)$, so that $\psi(x, 0)$ is real and even.

  Use the method of stationary phase to obtain an approximation to $\psi(x, t)$ for large $t$, with $x / t$ taking the constant value $V$, and $0 \leqslant V<c$. Explain briefly why your answer is inappropriate if $V>c$.

  [You are given that

  $$\left.\int_{-\infty}^{\infty} \exp \left(i u^{2}\right) d u=\pi^{1 / 2} e^{i \pi / 4} .\right]$$

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• Paper 2, Section II, C

  #Waves

  2013

  Show that the equations governing linear elasticity have plane-wave solutions, distinguishing between $\mathrm{P}, \mathrm{SV}$ and $\mathrm{SH}$ waves.

  A semi-infinite elastic medium in $y<0$ (where $y$ is the vertical coordinate) with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ is overlaid by a layer of thickness $h($ in $0<y<h)$ of a second elastic medium with density $\rho^{\prime}$ and Lamé moduli $\lambda^{\prime}$ and $\mu^{\prime}$. The top surface at $y=h$ is free, that is, the surface tractions vanish there. The speed of the S-waves is lower in the layer, that is, $c_{S}^{\prime 2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. For a time-harmonic SH-wave with horizontal wavenumber $k$ and frequency $\omega$, which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent horizontal wave speed $c(k)=\omega / k$ :

  $$\tan \left(k h \sqrt{\left(c^{2} / c_{S}^{\prime 2}\right)-1}\right)=\frac{\mu \sqrt{1-\left(c^{2} / c_{S}^{2}\right)}}{\mu^{\prime} \sqrt{\left(c^{2} / c_{S}^{\prime 2}\right)-1}}$$

  Show graphically that for a given value of $k$ there is always at least one real value of $c$ which satisfies equation $(*)$. Show further that there are one or more higher modes if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h .$

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• Paper 3, Section II, C

  #Waves

  2013

  The dispersion relation for sound waves of frequency $\omega$ in a stationary homogeneous gas is $\omega=c_{0}|\mathbf{k}|$, where $c_{0}$ is the speed of sound and $\mathbf{k}$ is the wavenumber. Derive the dispersion relation for sound waves of frequency $\omega$ in a uniform flow with velocity U.

  For a slowly-varying medium with local dispersion relation $\omega=\Omega(\mathbf{k}, \mathbf{x}, t)$, derive the ray-tracing equations

  $$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$$

  explaining carefully the meaning of the notation used.

  Suppose that two-dimensional sound waves with initial wavenumber $\left(k_{0}, l_{0}, 0\right)$ are generated at the origin in a gas occupying the half-space $y>0$. If the gas has a slowlyvarying mean velocity $(\gamma y, 0,0)$, where $\gamma>0$, show:

  (a) that if $k_{0}>0$ and $l_{0}>0$ the waves reach a maximum height (which should be identified), and then return to the level $y=0$ in a finite time;

  (b) that if $k_{0}<0$ and $l_{0}>0$ then there is no bound on the height to which the waves propagate.

  Comment briefly on the existence, or otherwise, of a quiet zone.

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• Paper 1, Section II, 39C

  #Waves

  2013

  Starting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants

  $$R_{\pm}=u \pm \frac{2}{\gamma-1}\left(c-c_{0}\right)$$

  are constant on characteristics $C_{\pm}$given by $d x / d t=u \pm c$, where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma$ is the ratio of specific heats.

  Such a gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest with $c=c_{0}$. At time $t=0$ the piston starts moving to the left at a constant velocity $V$. Find $u(x, t)$ and $c(x, t)$ in the three regions

  $$\begin{array}{cc} \text { (i) } & c_{0} t \leqslant x \\ \text { (ii) } & a t \leqslant x \leqslant c_{0} t \\ \text { (iii) } & -V t \leqslant x \leqslant a t, \end{array}$$

  where $a=c_{0}-\frac{1}{2}(\gamma+1) V$. What is the largest value of $V$ for which $c$ is positive throughout region (iii)? What happens if $V$ exceeds this value?

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• Paper 3, Section I, A

  #Variational Principles

  2013

  A cylindrical drinking cup has thin curved sides with density $\rho$ per unit area, and a disk-shaped base with density $k \rho$ per unit area. The cup has capacity to hold a fixed volume $V$ of liquid. Use the method of Lagrange multipliers to find the minimum mass of the cup.

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• Paper 2, Section II, A

  #Variational Principles

  2013

  Starting from the Euler-Lagrange equation, show that a condition for

  $$\int f\left(y, y^{\prime}\right) d x$$

  to be stationary is

  $$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant }$$

  In the half-plane $y>0$, light has speed $c(y)=y+c_{0}$ where $c_{0}>0$. Find the equation for a light ray between $(-a, 0)$ and $(a, 0)$. Sketch the solution.

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• Paper 4, Section II, A

  #Variational Principles

  2013

  Derive the Euler-Lagrange equation for the integral

  $$\int_{a}^{b} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x$$

  where prime denotes differentiation with respect to $x$, and both $y$ and $y^{\prime}$ are specified at $x=a, b$.

  Find $y(x)$ that extremises the integral

  $$\int_{0}^{\pi}\left(y+\frac{1}{2} y^{2}-\frac{1}{2} y^{\prime \prime 2}\right) d x$$

  subject to $y(0)=-1, y^{\prime}(0)=0, y(\pi)=\cosh \pi$ and $y^{\prime}(\pi)=\sinh \pi$.

  Show that your solution is a global maximum. You may use the result that

  $$\int_{0}^{\pi} \phi^{2}(x) d x \leqslant \int_{0}^{\pi} \phi^{\prime 2}(x) d x$$

  for any (suitably differentiable) function $\phi$ which satisfies $\phi(0)=0$ and $\phi(\pi)=0$.

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